Fractional Jensen-Mercer Type Inequalities Involving Generalized Raina’s Function and Applications

Abstract

The aim of this paper is to derive some new generalized fractional analogues of Mercer type inequalities, essentially using the convexity property of the functions and Raina’s function. We also discuss several new special cases which show that our results are, to an extent, unifying. In order to illustrate the significance of our results, we offer some interesting applications of our results to special means, error bounds, and q-digamma functions.

1. Introduction

Fractional Calculus (FC) can be viewed as a generalization of classical calculus with respect to the operations of integration and differentiation of any arbitrary order. The history of fractional calculus is almost simultaneous with ordinary calculus. As compared to the ordinary counterparts in many physical problems, the concepts of fractional calculus have proven to be more useful. Consequently, in the past few decades, it has experienced a rapid development. Many classical concepts of fractional calculus have been extended and generalized according to need. One of the most significant concepts belonging to fractional calculus is Riemann-Liouville’s fractional integrals.

Definition 1

([1]). Let $\mathsf{\Phi }\in L[{\upsilon }_1,{\upsilon }_2]$. The Riemann-Liouville integrals $J_{{\upsilon }_1+}^{\lambda }\mathsf{\Phi }$ and $J_{{\upsilon }_2-}^{\lambda }\mathsf{\Phi }$ of order $\lambda >0$ with ${\upsilon }_1\ge 0$ are defined by

$$\begin{array}{c}\hfill J_{{\upsilon }_1+}^{\lambda }\mathsf{\Phi }\left(u_1\right)=\frac{1}{\mathsf{\Gamma }\left(\lambda \right)}{\int }_{{\upsilon }_1}^{u_1}{(u_1-\delta )}^{\lambda -1}\mathsf{\Phi }\left(\delta \right)d\delta ,(u_1>{\upsilon }_1),\end{array}$$

and

$$\begin{array}{c}\hfill J_{{\upsilon }_2-}^{\lambda }\mathsf{\Phi }\left(u_1\right)=\frac{1}{\mathsf{\Gamma }\left(\lambda \right)}{\int }_{u_1}^{{\upsilon }_2}{(\delta -u_1)}^{\lambda -1}\mathsf{\Phi }\left(\delta \right)d\delta ,(u_1<{\upsilon }_2),\end{array}$$

respectively, where $\mathsf{\Gamma }\left(\lambda \right)={\int }_0^{\infty }{e}^{-\delta }{\delta }^{\lambda -1}\mathrm{d}\delta$. Here is $J_{{\upsilon }_1+}^0\mathsf{\Phi }\left(u_1\right)=J_{{\upsilon }_2-}^0\mathsf{\Phi }\left(u_1\right)=\mathsf{\Phi }\left(u_1\right).$

Mubeen and Habibullah [2] introduced the class of k-Riemann-Liouville fractional integrals and discussed its basic properties. They have observed that the properties of k-Riemann-Liouville fractional integrals are quite different. Thus, we can say that k-Riemann-Liouville fractional integrals are significant generalizations of Riemann-Liouville fractional integrals.

Definition 2

([2]). Let $\mathsf{\Phi }\in L[{\upsilon }_1,{\upsilon }_2]$. Then for $k>0,$ the left and right k-Riemann-Liouville fractional integrals are defined as:

$$\begin{array}{c}\hfill {}_k{I}_{{{\upsilon }_1}^{+}}^{\lambda }\mathsf{\Phi }\left(u_1\right)=\frac{1}{k{\mathsf{\Gamma }}_k\left(\lambda \right)}{\int }_{{\upsilon }_1}^{u_1}{(u_1-\delta )}^{\frac{\lambda }{k}-1}\mathsf{\Phi }\left(\delta \right)d\delta ,(u_1>{\upsilon }_1),\end{array}$$

and

$$\begin{array}{c}\hfill {}_k{I}_{{{\upsilon }_2}^{-}}^{\lambda }\mathsf{\Phi }\left(u_1\right)=\frac{1}{k{\mathsf{\Gamma }}_k\left(\lambda \right)}{\int }_{u_1}^{{\upsilon }_2}{(\delta -u_1)}^{\frac{\lambda }{k}-1}\mathsf{\Phi }\left(\delta \right)d\delta ,(u_1<{\upsilon }_2),\end{array}$$

where ${\mathsf{\Gamma }}_k(·)$ is the k-gamma function.

Another fractional integral defined in the literature is $\psi$-Riemann-Liouville fractional integrals.

Definition 3

([1]). Let $\psi :[{\upsilon }_1,{\upsilon }_2]\to {\mathbb{R}}_{+}$ be an increasing function possessing continuous derivative ${\psi }^{\prime }$. Then, for $\lambda >0,$ the left and right ψ-Riemann-Liouville fractional integrals of Φ with respect to ψ are defined as:

$$\begin{array}{c}\hfill I_{{\upsilon }_1+}^{\lambda ,\psi }\mathsf{\Phi }\left(u_1\right)=\frac{1}{\mathsf{\Gamma }\left(\lambda \right)}{\int }_{{\upsilon }_1}^{u_1}{(\psi \left(u_1\right)-\psi \left(\delta \right))}^{\lambda -1}{\psi }^{\prime }\left(\delta \right)\mathsf{\Phi }\left(\delta \right)d\delta ,(u_1>{\upsilon }_1),\end{array}$$

and

$$\begin{array}{c}\hfill I_{{{\upsilon }_2}^{-}}^{\lambda ,\psi }\mathsf{\Phi }\left(u_1\right)=\frac{1}{\mathsf{\Gamma }\left(\lambda \right)}{\int }_{u_1}^{{\upsilon }_2}{(\psi \left(\delta \right)-\psi \left(u_1\right))}^{\lambda -1}{\psi }^{\prime }\left(\delta \right)\mathsf{\Phi }\left(\delta \right)d\delta ,(u_1<{\upsilon }_2).\end{array}$$

If we take $\psi \left(\delta \right)=\delta$ that is identity function in Definition 3, then we recapture the definition of Riemann-Liouville fractional integrals.

Akkurt et al. [3] extended the notions of Riemann-Liouville fractional integrals, k-Riemann-Liouville fractional integrals and $\psi$-Riemann-Liouville fractional integrals and defined the class of ${\psi }_k$-fractional integrals as:

Definition 4.

Let $\mathsf{\Phi }\in L[{\upsilon }_1,{\upsilon }_2]$ and $\psi :[{\upsilon }_1,{\upsilon }_2]\to {\mathbb{R}}_{+}$ be an increasing function possessing continuous derivative ${\psi }^{\prime }$. Then, for $k>0,$ the left and right ${\psi }_k$-Riemann-Liouville fractional integrals of Φ with respect to ψ are defined as

$$\begin{array}{c}\hfill {}_k{I}_{{\upsilon }_1+}^{\lambda ,\psi }\mathsf{\Phi }\left(u_1\right)=\frac{1}{k{\mathsf{\Gamma }}_k\left(\lambda \right)}{\int }_{{\upsilon }_1}^{u_1}{(\psi \left(u_1\right)-\psi \left(\delta \right))}^{\frac{\lambda }{k}-1}{\psi }^{\prime }\left(\delta \right)\mathsf{\Phi }\left(\delta \right)d\delta ,(u_1>{\upsilon }_1),\end{array}$$

and

$$\begin{array}{c}\hfill {}_k{I}_{{{\upsilon }_2}^{-}}^{\lambda ,\psi }\mathsf{\Phi }\left(u_1\right)=\frac{1}{k{\mathsf{\Gamma }}_k\left(\lambda \right)}{\int }_{u_1}^{{\upsilon }_2}{(\psi \left(\delta \right)-\psi \left(u_1\right))}^{\frac{\lambda }{k}-1}{\psi }^{\prime }\left(\delta \right)\mathsf{\Phi }\left(\delta \right)d\delta ,(u_1<{\upsilon }_2).\end{array}$$

Recently, Awan et al. [4] have obtained some new Hermite-Hadamard like inequalities using the Definition 4.

On the basis of k-gamma function Tunc et al. [5] introduced a new class of function (k-Raina’s function), which is defined as:

$$\begin{array}{c}\hfill {\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left(u_1\right)=\sum _{i=0}^{\infty }\frac{\varpi \left(i\right){u_1}^i}{k{\mathsf{\Gamma }}_k(\varrho ik+\lambda )}(\varrho ,\nu >0;|u_1|<\mathbb{R}),\end{array}$$

where coefficients $\varpi \left(i\right)$ with $(i\in \mathbb{N}\cup \{0\left\}\right)$ are bounded sequences of positive real numbers and $\mathbb{R}$ is a set of real numbers. With the help of these functions, Tunc et al. have defined the left and right-sided k-Raina’s fractional integrals, respectively, as:

$$\begin{array}{cc}\hfill & \left(J_{\varrho ,\lambda ,{{\upsilon }_1}^{+};\omega }^{\varpi ,k}\right)\left(u_1\right)={\int }_{{\upsilon }_1}^{u_1}{(u_1-\delta )}^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\nu }^{\varpi ,k}\left[\omega {(u_1-\delta )}^{\varrho }\right]\mathsf{\Phi }\left(\delta \right)\mathrm{d}\delta (u_1>{\upsilon }_1>0),\hfill \\ \hfill & \left(J_{\varrho ,\lambda ,{{\upsilon }_2}^{-};\omega }^{\varpi ,k}\right)\left(u_1\right)={\int }_{u_1}^{{\upsilon }_2}{(\delta -u_1)}^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\nu }^{\varpi ,k}\left[\omega {(\delta -u_1)}^{\varrho }\right]\mathsf{\Phi }\left(\delta \right)\mathrm{d}\delta (0<u_1<{\upsilon }_2),\hfill \end{array}$$

where $\varrho ,\lambda >0$ and $\omega \in \mathbb{R}$.

1.

For $k=1$, we obtain Raina’s fractional integral, see [6].

2.

If we set $k=1,i=0$ and $\varpi \left(0\right)=1$, then we obtain the classical Riemann-Liouville fractional integrals.

Set et al. [7] also introduced the following generalized fractional integrals:

Definition 5.

Let $\mathsf{\Phi }\in L[{\upsilon }_1,{\upsilon }_2]$ and $\psi :[{\upsilon }_1,{\upsilon }_2]\to {\mathbb{R}}_{+}$ be an increasing function possessing continuous derivative ${\psi }^{\prime }$. Then, for $k>0,$ the left and right ${\psi }_k$-Raina fractional integrals of Φ with respect to ψ are defined as:

$$\begin{array}{c}\hfill I_{\varrho ,\lambda ,{{\upsilon }_1}^{+};\omega }^{\varpi ,\psi ,k}\mathsf{\Phi }\left(u_1\right)={\int }_{{\upsilon }_1}^{u_1}{(\psi \left(u_1\right)-\psi \left(\delta \right))}^{\frac{\lambda }{k}-1}{\psi }^{\prime }\left(\delta \right){\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(\psi \left(u_1\right)-\psi \left(\delta \right))}^{\varrho }\right]\mathsf{\Phi }\left(\delta \right)d\delta ,\end{array}$$

for $u_1>{\upsilon }_1$ and

$$\begin{array}{c}\hfill I_{\varrho ,\lambda ,{{\upsilon }_2}^{-};\omega }^{\varpi ,\psi ,k}\mathsf{\Phi }\left(u_1\right)={\int }_{u_1}^{{\upsilon }_2}{(\psi \left(\delta \right)-\psi \left(u_1\right))}^{\frac{\lambda }{k}-1}{\psi }^{\prime }\left(\delta \right){\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(\psi \left(\delta \right)-\psi \left(u_1\right))}^{\varrho }\right]\mathsf{\Phi }\left(\delta \right)d\delta ,\end{array}$$

for $u_1<{\upsilon }_2$, where $\lambda ,\varrho >0$ and $\omega \in \mathbb{R}$.

If we take $k=1$ in Definition 5, then we have $\psi$-Raina’s fractional integral which is given as:

Definition 6.

Let $\mathsf{\Phi }\in L[{\upsilon }_1,{\upsilon }_2]$ and $\psi :[{\upsilon }_1,{\upsilon }_2]\to {\mathbb{R}}_{+}$ be an increasing function possessing continuous derivative ${\psi }^{\prime }$. Then left and right ψ-Raina fractional integrals of Φ with respect to ψ are defined as

$$\begin{array}{c}\hfill I_{\varrho ,\lambda ,{{\upsilon }_1}^{+};\omega }^{\varpi ,\psi }\mathsf{\Phi }\left(u_1\right)={\int }_{{\upsilon }_1}^{u_1}{(\psi \left(u_1\right)-\psi \left(\delta \right))}^{\lambda -1}{\psi }^{\prime }\left(\delta \right){\mathcal{F}}_{\varrho ,\lambda }^{\varpi }\left[\omega {(\psi \left(u_1\right)-\psi \left(\delta \right))}^{\varrho }\right]\mathsf{\Phi }\left(\delta \right)d\delta ,\end{array}$$

for $u_1>{\upsilon }_1$ and

$$\begin{array}{c}\hfill I_{\varrho ,\lambda ,{{\upsilon }_2}^{-};\omega }^{\varpi ,\psi }\mathsf{\Phi }\left(u_1\right)={\int }_{u_1}^{{\upsilon }_2}{(\psi \left(\delta \right)-\psi \left(u_1\right))}^{\lambda -1}{\psi }^{\prime }\left(\delta \right){\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,}\left[\omega {(\psi \left(\delta \right)-\psi \left(u_1\right))}^{\varrho }\right]\mathsf{\Phi }\left(\delta \right)d\delta ,\end{array}$$

for $u_1<{\upsilon }_2$, where $\lambda ,\varrho >0$ and $\omega \in \mathbb{R}$.

Remark 1.

1.

If we take $i=0$ and $\varpi \left(0\right)=1$ in Definition 5, then we have ${\psi }_k$-Riemann-Liouville fractional integrals.

2.

If we set $\psi \left(u_1\right)=u_1,i=0$ and $\varpi \left(0\right)=1$ in Definition 5, then we get k-Riemann-Liouville fractional integrals.

3.

If we choose $\psi \left(u_1\right)=u_1,i=0,k=1$ and $\varpi \left(0\right)=1$ in Definition 5, then we obtain Riemann-Liouville fractional integrals.

For further information, see [6].

The theory of convexity is one of the most significant sides of mathematical analysis in which we study the properties of convex sets and convex functions. These classical concepts have played a vital role in different branches of pure and applied sciences. Applications for integral inequalities involving convexity are numerous in a number of areas of mathematics where symmetry is crucial. The classical concept of convexity and symmetry are both connected with each other, so you may use one and apply it to the other.

A set $\mathcal{C}\subset \mathbb{R}$ is said to be convex, if

$$\begin{array}{c}\hfill (1-\delta ){\upsilon }_1+\delta {\upsilon }_2\in \mathcal{C},\forall {\upsilon }_1,{\upsilon }_2\in \mathcal{C},\delta \in [0,1].\end{array}$$

Similarly, a function $\mathsf{\Phi }:\mathcal{C}\to \mathbb{R}$ is said to be convex, if

$$\begin{array}{c}\hfill \mathsf{\Phi }((1-\delta ){\upsilon }_1+\delta {\upsilon }_2)\le (1-\delta )\mathsf{\Phi }\left({\upsilon }_1\right)+\delta \mathsf{\Phi }\left({\upsilon }_2\right),\forall {\upsilon }_1,{\upsilon }_2\in \mathcal{C},\delta \in [0,1].\end{array}$$

Another aspect of convexity theory that has made it more charming is its relation to the theory of inequalities. A huge number of inequalities known to us in the literature can be obtained by using the convexity property of the functions. One of the most studied results in this regard is Hermite-Hadamard’s inequality. For some interesting details, see [8,9,10,11,12,13]. This result provides us with a necessary and sufficient condition for a function to be convex. It reads as:

Let $\mathsf{\Phi }:\mathcal{I}\subseteq \mathbb{R}\to \mathbb{R}$ be a convex function, then

$$\begin{array}{c}\hfill \mathsf{\Phi }\left(\frac{{\upsilon }_1+{\upsilon }_2}{2}\right)\le \frac{1}{{\upsilon }_2-{\upsilon }_1}\underset{{\upsilon }_1}{\overset{{\upsilon }_2}{\int }}\mathsf{\Phi }\left(u_1\right)\mathrm{d}{u}_1\le \frac{\mathsf{\Phi }\left({\upsilon }_1\right)+\mathsf{\Phi }\left({\upsilon }_2\right)}{2}.\end{array}$$

Jensen’s inequality [14] is another significant pertaining to convexity property of the functions. Let $\mathsf{\Phi }$ be a convex function on $[{\upsilon }_1,{\upsilon }_2]$, then for all ${u_1}_i\in [{\upsilon }_1,{\upsilon }_2]$ and ${\mu }_i\in [0,1],$$i=1,2,\dots ,n$, with ${\sum }_{i=1}^n{\mu }_i=1$, we have

$$\begin{array}{c}\hfill \mathsf{\Phi }\left(\sum _{i=1}^n{\mu }_i{u_1}_i\right)\le \sum _{i=1}^n{\mu }_i\mathsf{\Phi }\left({u_1}_i\right).\end{array}$$

The following inequality is known in the literature as the Jensen-Mercer’s inequality [15]:

$$\begin{array}{c}\hfill \mathsf{\Phi }\left({\upsilon }_1+{\upsilon }_2-\sum _{i=1}^n{\mu }_i{u_1}_i\right)\le \mathsf{\Phi }\left({\upsilon }_1\right)+\mathsf{\Phi }\left({\upsilon }_2\right)-\sum _{i=1}^n{\mu }_i\mathsf{\Phi }\left({u_1}_i\right),\end{array}$$

whenever $\mathsf{\Phi }:[{\upsilon }_1,{\upsilon }_2]⟶\mathbb{R}$ is a convex function. For more details, see [15].

Pavić [16] obtained a generalized version of the Jensen-Mercer’s inequality in the following way:

Assume that $\mathsf{\Phi }:[{\upsilon }_1,{\upsilon }_2]\to \mathbb{R}$ is a convex function, where ${u_1}_i\in [{\upsilon }_1,{\upsilon }_2]$ are n-points. Let ${\varpi }_1,{\varpi }_2,{\mu }_i\in [0,1]$, ${\varpi }_3\in [-1,1]$ be coefficients such that ${\varpi }_1+{\varpi }_2+{\varpi }_3={\sum }_{i=1}^n{\mu }_i=1$, then

$$\begin{array}{c}\hfill \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\sum _{i=1}^n{\mu }_i{u_1}_i\right)\le {\varpi }_1\mathsf{\Phi }\left({\upsilon }_1\right)+{\varpi }_2\mathsf{\Phi }\left({\upsilon }_2\right)+{\varpi }_3\sum _{i=1}^n{\mu }_i\mathsf{\Phi }\left({u_1}_i\right).\end{array}$$

(1)

Remark 2.

Note that

1.

If we take ${\varpi }_1=1={\varpi }_2$ and ${\varpi }_3=-1$ in (1), then we have Jensen-Mercer inequality.

2.

If we set ${\varpi }_1=0={\varpi }_2$ and ${\varpi }_3=1$ in (1), then we get Jensen inequality.

3.

If we choose ${\varpi }_3=0$ in (1), then we obtain convex inequality.

For more details about Hermite-Hadamard-Mercer’s and Jensen-Mercer’s inequalities, see [17,18,19].

The main objective of this paper is to derive some new generalized fractional analogues of Mercer type inequalities essentially using the convexity property of the functions and with Raina’s function. We discuss several new special cases which show that our results are quite unifying. In order to illustrate the significance of our results, we also present some interesting applications of our results to special means, error bounds and q-digamma functions. Finally, some conclusions and future research in this fascinating field are given. We hope that the ideas of this paper will inspire interested readers working in this field.

2. Main Results

In this section, we discuss our main results.

Auxiliary Results

We now derive two new auxiliary results. These results will play an important role in the development of our results.

Lemma 1.

Let $\mathsf{\Phi }:[{\upsilon }_1,{\upsilon }_2]\to \mathbb{R}$ be a differentiable function on $({\upsilon }_1,{\upsilon }_2)$ with ${\upsilon }_1<{\upsilon }_2$. If ${\mathsf{\Phi }}^{\prime }\in L[{\upsilon }_1,{\upsilon }_2]$, then

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\frac{\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}{2}\hfill \\ \hfill & -\frac{1}{2k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right.\hfill \\ \hfill & +\left.I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right]\hfill \\ \hfill & =\frac{{\varpi }_3(u_2-u_1)}{2}\hfill \\ \hfill & \times \left[{\int }_0^1{(1-\delta )}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]{\mathsf{\Phi }}^{\prime }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2))d\delta \right.\hfill \\ \hfill & \left.-{\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]{\mathsf{\Phi }}^{\prime }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2))d\delta \right],\hfill \end{array}$$

for all $u_1,u_2\in [{\upsilon }_1,{\upsilon }_2],u_1<u_2$ and $\lambda ,\varrho ,\omega >0$.

Proof. 

Note that it is sufficient to check that $I:=\frac{{\varpi }_3(u_2-u_1)}{2}(I_2-I_1)$, where

$$\begin{array}{cc}\hfill & I_1={\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]{\mathsf{\Phi }}^{\prime }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2))d\delta \hfill \\ \hfill & ={\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]\frac{\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2))}{{\varpi }_3(u_1-u_2)}{|}_0^1\hfill \\ & -{\int }_0^1\frac{\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2))}{{\varpi }_3(u_1-u_2)}\frac{\mathrm{d}}{\mathrm{d}\delta }\left(\sum _{i=0}^{\infty }\frac{{\omega }^i{(u_2-u_1)}^{\varrho i}{\left({\varpi }_3\right)}^{\varrho i}{\delta }^{\frac{\lambda }{k}+\varrho i}}{k{\mathsf{\Gamma }}_k(\varrho ik+\lambda +k)}\right)d\delta \hfill \\ \hfill & =-\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)}{{\varpi }_3(u_2-u_1)}\hfill \\ & -{\int }_0^1\frac{\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2))}{{\varpi }_3(u_1-u_2)}\left(\frac{{\delta }^{\frac{\lambda }{k}-1}}{k}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]\right)d\delta \hfill \\ \hfill & =-\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)}{{\varpi }_3(u_2-u_1)}+\frac{1}{{\varpi }_{3}k(u_2-u_1)}\hfill \\ \hfill & \times {\int }_0^1{\delta }^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2)d\delta \hfill \\ \hfill & =-\frac{{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)}{{\varpi }_3(u_2-u_1)}\hfill \\ \hfill & +\frac{1}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}+1}}{I}_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right]}^{+},\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & I_2={\int }_0^1{(1-\delta )}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]{\mathsf{\Phi }}^{\prime }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2))d\delta \hfill \\ \hfill & ={(1-\delta )}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}[\omega {(u_2-u_1)}^{\varrho }\left({\varpi }_3{(1-\delta )}^{\varrho }\right]\frac{\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2))}{{\varpi }_3(u_1-u_2)}{|}_0^1\hfill \\ \hfill & -{\int }_0^1\frac{{(1-\delta )}^{\frac{\lambda }{k}-1}}{{\varpi }_{3}k(u_2-u_1)}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi }\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2))d\delta \hfill \\ \hfill & =\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}{{\varpi }_3(u_2-u_1)}\hfill \\ \hfill & -{\int }_0^1\frac{{(1-\delta )}^{\frac{\lambda }{k}-1}}{{\varpi }_{3}k(u_2-u_1)}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi }\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2))d\delta \hfill \\ \hfill & =\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}{{\varpi }_3(u_2-u_1)}-\frac{1}{{\varpi }_{3}k(u_2-u_1)}\hfill \\ \hfill & \times {\int }_0^1{(1-\delta )}^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi }\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2)d\delta \hfill \\ \hfill & =\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}{{\varpi }_3(u_2-u_1)}\hfill \\ \hfill & -\frac{1}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}+1}}{I}_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right]}^{-},\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right).\hfill \end{array}$$

Combining $I_1$ and $I_2$ with I, we get the required result. □

We now derive our second auxiliary result.

Lemma 2.

Let $\mathsf{\Phi }:[{\upsilon }_1,{\upsilon }_2]\to \mathbb{R}$ be a differentiable function on $({\upsilon }_1,{\upsilon }_2)$ with ${\upsilon }_1<{\upsilon }_2$. If ${\mathsf{\Phi }}^{\prime }\in L[{\upsilon }_1,{\upsilon }_2]$, then

$$\begin{array}{cc}\hfill & \frac{{(n+1)}^{\frac{\lambda }{k}-1}}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right.\hfill \\ \hfill & \left.+I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right]\hfill \\ \hfill & -\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{(n+1)}\hfill \\ \hfill & \times \left[\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right)\right)+\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\right]\hfill \\ \hfill & =\frac{{\varpi }_3(u_2-u_1)}{{(n+1)}^2}\{{\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ \hfill & \times {\mathsf{\Phi }}^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{\delta }{n+1}{u}_1+\frac{n+1-\delta }{n+1}{u}_2\right)\right)d\delta \hfill \\ \hfill & -{\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ \hfill & \times {\mathsf{\Phi }}^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2\right)\right)d\delta \},\hfill \end{array}$$

for all $u_1,u_2\in [{\upsilon }_1,{\upsilon }_2],u_1<u_2,\lambda ,\varrho ,\omega >0$ and $\delta \in [0,1]$.

Proof. 

Suppose that

$$\begin{array}{cc}\hfill & I=\{{\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times {\mathsf{\Phi }}^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{\delta }{n+1}{u}_1+\frac{n+1-\delta }{n+1}{u}_2\right)\right)d\delta \hfill \\ \hfill & -{\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times {\mathsf{\Phi }}^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2\right)\right)d\delta \}=I_1-I_2.\hfill \end{array}$$

Note that,

$$\begin{array}{cc}\hfill & I_1={\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times {\mathsf{\Phi }}^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{\delta }{n+1}{u}_1+\frac{n+1-\delta }{n+1}{u}_2\right)\right)d\delta \hfill \\ \hfill & ={\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\frac{\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\frac{\delta }{n+1}{u}_1+\frac{n+1-\delta }{n+1}{u}_2)\right)}{{\varpi }_3(\frac{u_1}{n+1}-\frac{u_2}{n+1})}{|}_0^1\hfill \\ \hfill & -{\int }_0^1\frac{{\delta }^{\frac{\lambda }{k}-1}}{k}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi }\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\frac{\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\frac{\delta }{n+1}{u}_1+\frac{n+1-\delta }{n+1}{u}_2)\right)}{{\varpi }_3(\frac{u_1}{n+1}-\frac{u_2}{n+1})}\hfill \\ \hfill & =-\frac{(n+1){\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{{\varpi }_3(u_2-u_1)}\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right)\right)\hfill \\ \hfill & +\frac{n+1}{{\varpi }_3(u_2-u_1)}{\int }_0^1\frac{{\delta }^{\frac{\lambda }{k}-1}}{k}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi }\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{\delta }{n+1}{u}_1+\frac{n+1-\delta }{n+1}{u}_2\right)\right).\hfill \end{array}$$

Substituting $\mathsf{\Psi }\left(u\right)={\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left((\frac{\delta }{n+1}{u}_1+\frac{n+1-\delta }{n+1}{u}_2)\right)$ and after some computations, we get

$$\begin{array}{cc}\hfill & I_1=-\frac{(n+1){\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{{\varpi }_3(u_2-u_1)}\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right)\right)\hfill \\ \hfill & +\frac{{(n+1)}^{\frac{\lambda }{k}+1}}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}+1}}\hfill \\ & \times I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & I_2={\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times {\mathsf{\Phi }}^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2\right)\right)d\delta \hfill \\ \hfill & ={\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\frac{\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2\right)\right)}{{\varpi }_3(\frac{u_2}{n+1}-\frac{u_1}{n+1})}{|}_0^1\hfill \\ \hfill & -{\int }_0^1\frac{{\delta }^{\frac{\lambda }{k}-1}}{k}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi }\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\frac{\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2\right)\right)}{{\varpi }_3(\frac{u_2}{n+1}-\frac{u_1}{n+1})}\hfill \\ \hfill & =\frac{(n+1){\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{{\varpi }_3(u_2-u_1)}\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\hfill \\ \hfill & -\frac{n+1}{{\varpi }_3(u_2-u_1)}{\int }_0^1\frac{{\delta }^{\frac{\lambda }{k}-1}}{k}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi }\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2\right)\right).\hfill \end{array}$$

By substituting $\mathsf{\Psi }\left(v\right)={\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left((\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2)\right)$ and after some computations, we get

$$\begin{array}{cc}\hfill & I_2=\frac{(n+1){\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{{\varpi }_3(u_2-u_1)}\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\hfill \\ \hfill & -\frac{{(n+1)}^{\frac{\lambda }{k}+1}}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}+1}}\hfill \\ & \times I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right).\hfill \end{array}$$

By using $I_1$ and $I_2$, it follows that

$$\begin{array}{cc}\hfill & I_1-I_2=\frac{{(n+1)}^{\frac{\lambda }{k}+1}}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}+1}}\hfill \\ & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right.\hfill \\ \hfill & \left.+I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right]\hfill \\ \hfill & -\frac{(n+1){\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{{\varpi }_3(u_2-u_1)}\hfill \\ \hfill & \times \left[\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right)\right)+\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\right].\hfill \end{array}$$

By multiplying $\frac{{\varpi }_3(u_2-u_1)}{{(n+1)}^2}$ both sides of the above equality, we get the required result. □

Here, we drive some new generalized fractional Mercer type inequalities.

Theorem 1.

Suppose that $\mathsf{\Phi }:[{\upsilon }_1,{\upsilon }_2]\to \mathbb{R}$ is a convex function, then

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+u_2}{2}\right)\right)\hfill \\ \hfill & \le \frac{1}{2k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]}\hfill \\ \hfill & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right.\hfill \\ & \left.+I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right]\hfill \\ \hfill & \le \frac{\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}{2}\hfill \\ \hfill & \le {\varpi }_1\mathsf{\Phi }\left({\upsilon }_1\right)+{\varpi }_2\mathsf{\Phi }\left({\upsilon }_2\right)+{\varpi }_3\left[\frac{\mathsf{\Phi }\left(u_1\right)+\mathsf{\Phi }\left(u_2\right)}{2}\right],\hfill \end{array}$$

for all $u_1,u_2\in [{\upsilon }_1,{\upsilon }_2]$, $u_1<u_2$ and $\lambda ,\varrho ,\omega >0$.

Proof. 

By using the convexity property of $\mathsf{\Phi }$, we have

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{{u_1}_1+{u_2}_1}{2}\right)\right)\hfill \\ & =\mathsf{\Phi }\left(\frac{{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u_1}_1+{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u_2}_1}{2}\right)\hfill \\ \hfill & \le \frac{1}{2}\left[\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u_1}_1)+\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u_2}_1)\right],\hfill \end{array}$$

for all ${u_1}_1,{u_2}_1\in [{\upsilon }_1,{\upsilon }_2]$. By changing the variables, we have

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+u_2}{2}\right)\right)\hfill \\ \hfill & \le \frac{1}{2}\left[\mathsf{\Phi }(\delta ({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+(1-\delta )({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2))\right.\hfill \\ \hfill & \left.+\mathsf{\Phi }((1-\delta )({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+\delta ({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2))\right].\hfill \end{array}$$

By multiplying both sides of above inequality by ${\delta }^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]$ and then integrating the resulting inequality with respect to $\delta$ on $[0,1]$, we obtain

$$\begin{array}{cc}\hfill & k{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+u_2}{2}\right)\right)\hfill \\ \hfill & \le \frac{1}{2}[{\int }_0^1{\delta }^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]\hfill \\ \hfill & \times \mathsf{\Phi }\left(\delta ({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+(1-\delta )({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)d\delta \hfill \\ \hfill & +{\int }_0^1{\delta }^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]\hfill \\ \hfill & \times \mathsf{\Phi }\left((1-\delta )({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+\delta ({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)d\delta ]\hfill \\ \hfill & =\frac{1}{2{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\hfill \\ \hfill & \times \left[{\int }_{{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)}^{{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}{\left[\mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)-\mathsf{\Psi }\left(u\right)\right]}^{\frac{\lambda }{k}-1}{\mathsf{\Psi }}^{\prime }\left(u\right)\right.\hfill \\ \hfill & \times {\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {\left(\mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)-\mathsf{\Psi }\left(u\right)\right)}^{\varrho }\right]\mathsf{\Phi }\circ \mathsf{\Psi }\left(u\right)du\hfill \\ \hfill & +{\int }_{{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)}^{{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}{\left[\mathsf{\Psi }\left(u\right)-\mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right]}^{\frac{\lambda }{k}-1}{\mathsf{\Psi }}^{\prime }\left(u\right)\hfill \\ \hfill & \left.\times {\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {\left(\mathsf{\Psi }\left(u\right)-\mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right)}^{\varrho }\right]\mathsf{\Phi }\circ \mathsf{\Psi }\left(u\right)du\right]\hfill \\ \hfill & =\frac{1}{2k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\hfill \\ \hfill & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right.\hfill \\ \hfill & \left.+I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right].\hfill \end{array}$$

On the other hand, again using the convexity property of $\mathsf{\Phi }$ and using the Jensen-Mercer inequality, we have

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left(\delta ({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+(1-\delta ({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2))\right)\hfill \\ \hfill & +\mathsf{\Phi }\left((1-\delta )({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+\delta ({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\hfill \\ \hfill & \le \mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\hfill \\ \hfill & \le 2[{\varpi }_1\mathsf{\Phi }\left({\upsilon }_1\right)+{\varpi }_2\mathsf{\Phi }\left({\upsilon }_2\right)]+{\varpi }_3[\mathsf{\Phi }\left(u_1\right)+\mathsf{\Phi }\left(u_2\right)].\hfill \end{array}$$

Multiplying both sides of above inequalities by ${\delta }^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]$ and then integrating the resulting inequality with respect to $\delta$ on $[0,1]$. This completes the proof. □

Theorem 2.

Let $\mathsf{\Phi }:[{\upsilon }_1,{\upsilon }_2]\to \mathbb{R}$ be a convex function, then

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+u_2}{2}\right)\right)\hfill \\ \hfill & \le \frac{{(n+1)}^{\frac{\lambda }{k}}}{2k{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]{\left({\varpi }_3(u_2-u_1)\right)}^{\frac{\lambda }{k}}}\hfill \\ \hfill & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right.\hfill \\ \hfill & +\left.I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right]\hfill \\ \hfill & \le {\varpi }_1\mathsf{\Phi }\left({\upsilon }_1\right)+{\varpi }_2\mathsf{\Phi }\left({\upsilon }_2\right)+{\varpi }_3\left(\frac{\mathsf{\Phi }\left(u_1\right)+\mathsf{\Phi }\left(u_2\right)}{2}\right),\hfill \end{array}$$

for all $u_1,u_2\in [{\upsilon }_1,{\upsilon }_2]$, $u_1<u_2$ and $\lambda ,\varrho ,\omega >0$.

Proof. 

By using the convexity property of $\mathsf{\Phi }$, we conclude that

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{{u_1}_1+{u_2}_1}{2}\right)\right)\hfill \\ & \le \frac{1}{2}\left[\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u_1}_1)+\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u_2}_1)\right]\hfill \end{array}$$

By setting ${u_1}_1=\frac{\delta }{n+1}{u}_1+\frac{n+1-\delta }{n+1}{u}_2$ and ${u_2}_1=\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2$ for $u_1,u_2\in [{\upsilon }_1,{\upsilon }_2]$ and $\delta \in [0,1]$ in the above inequality, we find

$$\begin{array}{cc}\hfill & 2\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+u_2}{2}\right)\right)\hfill \\ \hfill & \le \left[\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{\delta }{n+1}{u}_1+\frac{n+1-\delta }{n+1}{u}_2\right)\right)\right.\hfill \\ \hfill & \left.+\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2\right)\right)\right].\hfill \end{array}$$

By multiplying both sides of above inequality by ${\delta }^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]$ and then integrating the resulting inequality with respect to $\delta$ on $[0,1]$, we have

$$\begin{array}{cc}\hfill & {\int }_0^{1}2{\delta }^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+u_2}{2}\right)\right)d\delta \hfill \\ \hfill & \le {\int }_0^1{\delta }^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{\delta }{n+1}{u}_1+\frac{n+1-\delta }{n+1}{u}_2\right)\right)d\delta \hfill \\ \hfill & +{\int }_0^1{\delta }^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2\right)\right)d\delta .\hfill \end{array}$$

This implies

$$\begin{array}{cc}\hfill & 2k{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]\hfill \\ & \times \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+u_2}{2}\right)\right)\hfill \\ \hfill & \le {\int }_0^1{\delta }^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{\delta }{n+1}{u}_1+\frac{n+1-\delta }{n+1}{u}_2\right)\right)d\delta \hfill \\ \hfill & +{\int }_0^1{\delta }^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2\right)\right)d\delta =\frac{{(n+1)}^{\frac{\lambda }{k}}}{{\left({\varpi }_3(u_2-u_1)\right)}^{\frac{\lambda }{k}}}\hfill \\ \hfill & \times {\int }_{{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))}^{{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}{\left[\mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)-\mathsf{\Psi }\left(u\right)\right]}^{\frac{\lambda }{k}-1}{\mathsf{\Psi }}^{\prime }\left(u\right)\hfill \\ \hfill & \times {\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {\left(\mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)-\mathsf{\Psi }\left(u\right)\right)}^{\varrho }\right]\mathsf{\Phi }\circ \mathsf{\Psi }\left(u\right)du\hfill \\ \hfill & +\frac{{(n+1)}^{\frac{\lambda }{k}}}{{\left({\varpi }_3(u_2-u_1)\right)}^{\frac{\lambda }{k}}}\hfill \\ & \times {\int }_{{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)}^{{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))}{\left[\mathsf{\Psi }\left(u\right)-\mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right]}^{\frac{\lambda }{k}-1}{\mathsf{\Psi }}^{\prime }\left(u\right)\hfill \\ \hfill & \times {\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {\left(\mathsf{\Psi }\left(u\right)-\mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right)}^{\varrho }\right]\mathsf{\Phi }\circ \mathsf{\Psi }\left(u\right)du\hfill \\ \hfill & =\frac{{(n+1)}^{\frac{\lambda }{k}}}{{\left({\varpi }_3(u_2-u_1)\right)}^{\frac{\lambda }{k}}}\hfill \\ & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right.\hfill \\ \hfill & +\left.I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right].\hfill \end{array}$$

This implies

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+u_2}{2}\right)\right)\hfill \\ \hfill & \le \frac{{(n+1)}^{\frac{\lambda }{k}}}{2k{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]{\left({\varpi }_3(u_2-u_1)\right)}^{\frac{\lambda }{k}}}\hfill \\ & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right.\hfill \\ \hfill & +\left.I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right].\hfill \end{array}$$

Again using the convexity property of the functions and the Jensen-Mercer inequality, we obtain

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{\delta }{n+1}{u}_1+\frac{n+1-\delta }{n+1}{u}_2\right)\right)\hfill \\ & +\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2\right)\right)\hfill \\ \hfill & \le 2\left[{\varpi }_1\mathsf{\Phi }\left({\upsilon }_1\right)+{\varpi }_2\mathsf{\Phi }\left({\upsilon }_2\right)\right]+{\varpi }_3(\mathsf{\Phi }\left(u_1\right)+\mathsf{\Phi }\left(u_2\right)).\hfill \end{array}$$

Multiplying both sides of the above inequality by ${\delta }^{\frac{\lambda }{k}-1}{\mathcal{F}}_{\varrho ,\lambda }^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]$ and then integrating the resulting inequality with respect to $\delta$ on $[0,1]$ completes the proof. □

Using Lemmas 1 and 2, we can derive the following next results.

Theorem 3.

Under the assumptions of Lemma 1, if $|{\mathsf{\Phi }}^{\prime }|$ is convex on $[{\upsilon }_1,{\upsilon }_2]$, then

$$\begin{array}{cc}\hfill & |{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\frac{\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}{2}\hfill \\ & -\frac{1}{2k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\hfill \\ \hfill & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right.\hfill \\ \hfill & \left.+I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right]|\hfill \\ \hfill & \le |{\varpi }_3|(u_2-u_1){\mathcal{F}}_{\varrho ,\lambda +2k}^{{\varpi }_0,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\hfill \\ & \times \left[{\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|+{\varpi }_3\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+|{\mathsf{\Phi }}^{\prime }\left(u_2\right)|}{2}\right)\right],\hfill \end{array}$$

where ${\varpi }_0\left(i\right):=\varpi \left(i\right)\left(1-\frac{1}{2^{\frac{\lambda }{k}}+\varrho i}\right)$ and for all $u_1,u_2\in [{\upsilon }_1,{\upsilon }_2]$ with $u_1<u_2$ and $\lambda ,\varrho ,\omega >0$.

Proof. 

By using Lemma 1 and the Jensen-Mercer inequality, we have

$$\begin{array}{cc}\hfill & |{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\frac{\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}{2}\hfill \\ & -\frac{1}{2k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\hfill \\ & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right.\hfill \\ \hfill & \left.+I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right]|\hfill \\ \hfill & \le \frac{|{\varpi }_3|(u_2-u_1)}{2}\hfill \\ & \times {\int }_0^1\left|{(1-\delta )}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]-{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]\right|\hfill \\ \hfill & \times \left|{\mathsf{\Phi }}^{\prime }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3(\delta u_1+(1-\delta )u_2))\right|d\delta \hfill \\ \hfill & \le \frac{|{\varpi }_3|(u_2-u_1)}{2}\hfill \\ & \times {\int }_0^1\left|{(1-\delta )}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]-{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]\right|\hfill \\ \hfill & \times \left[{\varpi }_1\right|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)|+{\varpi }_3\delta |{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+{\varpi }_3(1-\delta )|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\left|\right]d\delta \hfill \\ \hfill & =\frac{|{\varpi }_3|(u_2-u_1)}{2}\hfill \\ & \times {\int }_0^{\frac{1}{2}}{(1-\delta )}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]-{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]\hfill \\ \hfill & \times \left[{\varpi }_1\right|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)|+{\varpi }_3\delta |{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+{\varpi }_3(1-\delta )|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\left|\right]d\delta \hfill \\ \hfill & +\frac{{\varpi }_3(u_2-u_1)}{2}\hfill \\ & \times {\int }_{\frac{1}{2}}^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]-{(1-\delta )}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]\hfill \\ \hfill & \times \left[{\varpi }_1\right|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)|+{\varpi }_3\delta |{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+{\varpi }_3(1-\delta )|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\left|\right]d\delta \hfill \\ & =\frac{{\varpi }_3(u_2-u_1)}{2}(L_1+L_2).\hfill \end{array}$$

By calculating the constants $L_1$ and $L_2$, we obtain

$$\begin{array}{cc}\hfill & L_1=({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\left|\right)\hfill \\ \hfill & \times \left({\int }_0^{\frac{1}{2}}[{(1-\delta )}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]-{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]]d\delta \right)\hfill \\ \hfill & +{\varpi }_3\left|{\mathsf{\Phi }}^{\prime }\left(u_1\right)\right|\left({\int }_0^{\frac{1}{2}}\delta {(1-\delta )}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]d\delta \right.\hfill \\ \hfill & -\left.{\int }_0^{\frac{1}{2}}{\delta }^{\frac{\lambda }{k}+1}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]d\delta \right)\hfill \\ \hfill & +{\varpi }_3\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|\left({\int }_0^{\frac{1}{2}}{(1-\delta )}^{\frac{\lambda }{k}+1}{\mathcal{F}}_{\varrho ,{\varpi }_1+k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]d\delta \right.\hfill \\ \hfill & -\left.{\int }_0^{\frac{1}{2}}(1-\delta ){\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]d\delta \right)\hfill \\ \hfill & =({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\left|\right)\left({\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_1,k}\left[\omega {(u_2-u_1)}^{\varrho }{{\varpi }_3}^{\varrho }\right]-{\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_2,k}\left[\omega {(u_2-u_1)}^{\varrho }{{\varpi }_3}^{\varrho }\right]\right)\hfill \\ \hfill & +{\varpi }_3\left|{\mathsf{\Phi }}^{\prime }\left(u_1\right)\right|\left({\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_3,k}\left[\omega {(u_2-u_1)}^{\varrho }{{\varpi }_3}^{\varrho }\right]-{\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_4,k}\left[\omega {(u_2-u_1)}^{\varrho }{{\varpi }_3}^{\varrho }\right]\right)\hfill \\ \hfill & +{\varpi }_3\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|\left({\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_5,k}\left[\omega {(u_2-u_1)}^{\varrho }{{\varpi }_3}^{\varrho }\right]-{\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_6,k}\left[\omega {(u_2-u_1)}^{\varrho }{{\varpi }_3}^{\varrho }\right]\right).\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill L_1& =({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\left|\right)\left(\frac{1}{\frac{\lambda }{k}+\varrho i+1}-\frac{1}{2^{\frac{\lambda }{k}+\varrho i}(\frac{\lambda }{k}+\varrho i+1)}\right)\hfill \\ \hfill & +{\varpi }_3\left|{\mathsf{\Phi }}^{\prime }\left(u_1\right)\right|\left(\frac{1}{(\frac{\lambda }{k}+\varrho i+1)(\frac{\lambda }{k}+\varrho i+2)}-\frac{1}{2^{\frac{\lambda }{k}+\varrho i+1}(\frac{\lambda }{k}+\varrho i+1)}\right)\hfill \\ \hfill & +{\varpi }_3\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|\left(\frac{1}{\frac{\lambda }{k}+\varrho i+2}-\frac{1}{2^{\frac{\lambda }{k}+\varrho i+1}(\frac{\lambda }{k}+\varrho i+1)}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & L_2=({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\left|\right)\hfill \\ \hfill & \times \left({\int }_{\frac{1}{2}}^1\left[{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]-{(1-\delta )}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]\right]d\delta \right)\hfill \\ \hfill & +{\varpi }_3\left|{\mathsf{\Phi }}^{\prime }\left(u_1\right)\right|\left({\int }_{\frac{1}{2}}^1{\delta }^{\frac{\lambda }{k}+1}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]d\delta \right.\hfill \\ \hfill & -\left.{\int }_{\frac{1}{2}}^1\delta {(1-\delta )}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]d\delta \right)\hfill \\ \hfill & +{\varpi }_3\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|\left({\int }_{\frac{1}{2}}^1(1-\delta ){\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\delta \right)}^{\varrho }\right]d\delta \right.\hfill \\ \hfill & -\left.{\int }_{\frac{1}{2}}^1{(1-\delta )}^{\frac{\lambda }{k}+1}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3(1-\delta )\right)}^{\varrho }\right]d\delta \right)\hfill \\ \hfill & =({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\left|\right)\left({\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_1,k}\left[\omega {(u_2-u_1)}^{\varrho }{{\varpi }_3}^{\varrho }\right]-{\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_2,k}\left[\omega {(u_2-u_1)}^{\varrho }{{\varpi }_3}^{\varrho }\right]\right)\hfill \\ \hfill & +{\varpi }_3\left|{\mathsf{\Phi }}^{\prime }\left(u_1\right)\right|\left({\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_5,k}\left[\omega {(u_2-u_1)}^{\varrho }{{\varpi }_3}^{\varrho }\right]-{\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_6,k}\left[\omega {(u_2-u_1)}^{\varrho }{{\varpi }_3}^{\varrho }\right]\right)\hfill \\ \hfill & +{\varpi }_3\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|\left({\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_3,k}\left[\omega {(u_2-u_1)}^{\varrho }{{\varpi }_3}^{\varrho }\right]-{\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_4,k}\left[\omega {(u_2-u_1)}^{\varrho }{{\varpi }_3}^{\varrho }\right]\right)\hfill \\ \hfill & =({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\left|\right)\left(\frac{1}{\frac{\lambda }{k}+\varrho i+1}-\frac{1}{2^{\frac{\lambda }{k}+\varrho i}(\frac{\lambda }{k}+\varrho i+1)}\right)\hfill \\ \hfill & +{\varpi }_3\left|{\mathsf{\Phi }}^{\prime }\left(u_1\right)\right|\left(\frac{1}{\frac{\lambda }{k}+\varrho i+2}-\frac{1}{2^{\frac{\lambda }{k}+\varrho i+1}(\frac{\lambda }{k}+\varrho i+1)}\right)\hfill \\ \hfill & +{\varpi }_3\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|\left(\frac{1}{(\frac{\lambda }{k}+\varrho i+1)(\frac{\lambda }{k}+\varrho i+2)}-\frac{1}{2^{\frac{\lambda }{k}+\varrho i+1}(\frac{\lambda }{k}+\varrho i+1)}\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\varpi }_1\left(k\right)& :=\varpi \left(k\right)\left(\frac{1}{\frac{\lambda }{k}+\varrho i+1}-\frac{1}{2^{\frac{\lambda }{k}+\varrho i+1}(\frac{\lambda }{k}+\varrho i+1)}\right),\hfill \\ \hfill {\varpi }_2\left(k\right)& :=\varpi \left(k\right)\left(\frac{1}{2^{\frac{\lambda }{k}+\varrho i+1}(\frac{\lambda }{k}+\varrho i+1)}\right),\hfill \\ \hfill {\varpi }_3\left(k\right)& :=\varpi \left(k\right)\left(\frac{1}{\frac{\lambda }{k}+\varrho i+1}-\frac{1}{2^{\frac{\lambda }{k}+\varrho i+1}(\frac{\lambda }{k}+\varrho i+1)}+\frac{1}{2^{\frac{\lambda }{k}+\varrho i+2}(\frac{\lambda }{k}+\varrho i+2)}-\frac{1}{\frac{\lambda }{k}+\varrho i+2}\right),\hfill \\ \hfill {\varpi }_4\left(k\right)& :=\varpi \left(k\right)\left(\frac{1}{2^{\frac{\lambda }{k}+\varrho i+2}(\frac{\lambda }{k}+\varrho i+2)}\right),\hfill \\ \hfill {\varpi }_5\left(k\right)& :=\varpi \left(k\right)\left(\frac{1}{\frac{\lambda }{k}+\varrho i+2}-\frac{1}{2^{\frac{\lambda }{k}+\varrho i+2}(\frac{\lambda }{k}+\varrho i+2)}\right),\hfill \\ \hfill {\varpi }_6\left(k\right)& :=\varpi \left(k\right)\left(\frac{1}{2^{\frac{\lambda }{k}+\varrho i+1}(\frac{\lambda }{k}+\varrho i+1)}-\frac{1}{2^{\frac{\lambda }{k}+\varrho i+2}(\frac{\lambda }{k}+\varrho i+2)}\right).\hfill \end{array}$$

By substituting $L_1$ and $L_2$, we obtain the desired inequality. □

Theorem 4.

Under the all assumptions of Lemma 2, if $|{\mathsf{\Phi }}^{\prime }|$ is convex on $[{\upsilon }_1,{\upsilon }_2]$, then

$$\begin{array}{cc}\hfill & |\frac{{(n+1)}^{\frac{\lambda }{k}-1}}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\hfill \\ & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right.\hfill \\ \hfill & +\left.I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right]\hfill \\ \hfill & -\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{n+1}\hfill \\ \hfill & \times [\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right)\right)\hfill \\ & +\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\left]\right|\hfill \\ \hfill & \le \frac{2k\left|\varpi 3\right|(u_2-u_1){\mathcal{F}}_{\varrho ,\lambda +2k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{{(n+1)}^2}\hfill \\ & \left[{\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|+{\varpi }_3\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+|{\mathsf{\Phi }}^{\prime }\left(u_2\right)|}{2}\right)\right],\hfill \end{array}$$

for all $u_1,u_2\in [{\upsilon }_1,{\upsilon }_2],u_1<u_2,\lambda ,\varrho ,\omega >0,$ and $n\in \mathbb{N}$.

Proof. 

By using the Lemma 2 and Jensen-Mercer inequality, we have

$$\begin{array}{cc}\hfill & \frac{{(n+1)}^{\frac{\lambda }{k}-1}}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\hfill \\ & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right.\hfill \\ \hfill & \left.+I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right]\hfill \\ \hfill & -\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{n+1}\hfill \\ & \times \left[\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right)\right)+\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\right]|\hfill \\ \hfill & \le \frac{|{\varpi }_3\left(\right|u_2-u_1)}{{\left(n+1\right)}^2}\{{\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times \left|{\mathsf{\Phi }}^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{\delta u_1}{n+1}+\frac{n+1-\delta }{n+1}{u}_2\right)\right)d\delta \right|\hfill \\ \hfill & +{\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ & \times \left|{\mathsf{\Phi }}^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta }{n+1}{u}_2\right)\right)\right|d\delta \}\hfill \\ & \le \frac{|{\varpi }_3|(u_2-u_1)}{{\left(n+1\right)}^2}\left\{{\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\right.\hfill \\ \hfill & \times {\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)|+{\varpi }_3\left(\frac{\delta }{n+1}\right)|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+{\varpi }_3\left(\frac{n+1-\delta }{n+1}\right)\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|\left]d\delta \right.\hfill \\ \hfill & +{\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\hfill \\ \hfill & \times \left.[{\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)|+{\varpi }_3\left(\frac{n+1-\delta }{n+1}\right)|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+{\varpi }_3\left(\frac{\delta }{n+1}\right)|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\left|\right]d\delta \right\}\hfill \\ \hfill & \le \frac{|{\varpi }_3|(u_2-u_1)}{{\left(n+1\right)}^2}\left\{{\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\right.\hfill \\ \hfill & \times \left.[2{\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+2{\varpi }_2|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)|+{\varpi }_3\left(\right|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\left|\right)]d\delta \right\}\hfill \\ \hfill & \le \frac{|{\varpi }_3|(u_2-u_1)}{2{(n+1)}^2}\left\{{\int }_0^1{\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\right.\hfill \\ \hfill & \times \left.\left[{\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|+{\varpi }_3\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+|{\mathsf{\Phi }}^{\prime }\left(u_2\right)|}{2}\right)\right]d\delta \right\}\hfill \\ \hfill & \le \frac{2k|{\varpi }_3|(u_2-u_1){\mathcal{F}}_{\varrho ,\lambda +2k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{{(n+1)}^2}\hfill \\ & \times \left[{\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|+{\varpi }_3\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+|{\mathsf{\Phi }}^{\prime }\left(u_2\right)|}{2}\right)\right].\hfill \end{array}$$

This completes the proof. □

Theorem 5.

Under the assumptions of Lemma 2, if $|{\mathsf{\Phi }}^{\prime }{|}^q$ is convex on $[{\upsilon }_1,{\upsilon }_2]$ with $q>1$, then

$$\begin{array}{cc}\hfill & |\frac{{(n+1)}^{\frac{\lambda }{k}-1}}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right.\hfill \\ \hfill & \left.+I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right]\hfill \\ & -\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{n+1}\hfill \\ \hfill & \times \left[\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right)\right)+\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\right]|\hfill \\ \hfill & \le \frac{|{\varpi }_3|(u_2-u_1)}{{(n+1)}^2}{\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_1,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]\hfill \\ \hfill & \times \left[{\left({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right){|}^q+{\varpi }_2{\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|}^q+{\varpi }_3\left(\frac{(2n+1)|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+{\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q}{2(n+1)}\right)\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\left({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right){|}^q+{\varpi }_2{\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|}^q+{\varpi }_3\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+(2n+1){\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q}{2(n+1)}\right)\right)}^{\frac{1}{q}}],\hfill \end{array}$$

where ${\varpi }_1\left(i\right):=\varpi \left(i\right){\left(\frac{k}{\varrho pki+p\lambda +k}\right)}^{\frac{1}{p}}$ and for all $u_1,u_2\in [{\upsilon }_1,{\upsilon }_2]$ with $u_1<u_2$, $\lambda ,\varrho ,\omega >0$ and $\frac{1}{p}+\frac{1}{q}=1$.

Proof. 

By using Lemma 2 and the Hölder’s inequality, we get

$$\begin{array}{cc}\hfill & |\frac{{(n+1)}^{\frac{\lambda }{k}-1}}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right.\hfill \\ \hfill & \left.+I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right]\hfill \\ & -\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{n+1}\hfill \\ \hfill & \times \left[\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right)\right)+\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\right]|\hfill \\ \hfill & \le \frac{|{\varpi }_3|(u_2-u_1)}{{(n+1)}^2}{\left({\int }_0^1{\left({\delta }^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3\delta }{n+1}\right)}^{\varrho }\right]\right)}^{p}d\delta \right)}^{\frac{1}{p}}\hfill \\ \hfill & \times \left[{\left({\int }_0^1{\left|{\mathsf{\Phi }}^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta u_2}{n+1}\right)\right)\right|}^{q}d\delta \right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left({\int }_0^1{\left|{\mathsf{\Phi }}^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{\delta u_1}{n+1}+\frac{n+1-\delta }{n+1}{u}_2\right)\right)\right|}^{q}d\delta \right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & =\frac{|{\varpi }_3|(u_2-u_1)}{{(n+1)}^2}\sum _{i=1}^{\infty }\frac{{\varpi }_k\left(i\right){\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}^i}{k{\mathsf{\Gamma }}_k(\varrho ki+\lambda +k)}{\left({\int }_0^1{\delta }^{(\varrho i+\frac{\lambda }{k})p}\mathrm{d}\delta \right)}^{\frac{1}{p}}\hfill \\ \hfill & \times \left[{\left({\int }_0^1{\left|{\mathsf{\Phi }}^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n+1-\delta }{n+1}{u}_1+\frac{\delta u_2}{n+1}\right)\right)\right|}^{q}d\delta \right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left({\int }_0^1{\left|{\mathsf{\Phi }}^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{\delta u_1}{n+1}+\frac{n+1-\delta }{n+1}{u}_2\right)\right)\right|}^{q}d\delta \right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

By using the Jensen-Mercer inequality and taking into account the convexity of $|{\mathsf{\Phi }}^{\prime }{|}^q$, we find

$$\begin{array}{cc}\hfill & |\frac{{(n+1)}^{\frac{\lambda }{k}-1}}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\left[I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right)\right.\hfill \\ \hfill & \left.+I_{{\left[{\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right)\right]\hfill \\ & -\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{n+1}\hfill \\ \hfill & \times \left[\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right)\right)+\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\right]|\hfill \\ \hfill & \le \frac{|{\varpi }_3|(u_2-u_1)}{{(n+1)}^2}\sum _{i=1}^{\infty }\frac{{\varpi }_k\left(i\right){\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}^i}{k{\mathsf{\Gamma }}_k(\varrho ki+\lambda +k)}{\left(\frac{k}{\varrho pki+p\lambda +k}\right)}^{\frac{1}{p}}\hfill \\ \hfill & \times \left[{\left({\int }_0^1{\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right){|}^q+{\varpi }_2{\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|}^q+{\varpi }_3\left(\frac{n+1-\delta }{n+1}|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+\frac{\delta }{n+1}{\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q\right)d\delta \right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left({\int }_0^1{\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right){|}^q+{\varpi }_2{\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|}^q+{\varpi }_3\left(\frac{\delta }{n+1}|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+\frac{n+1-\delta }{n+1}{\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q\right)d\delta \right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & =\frac{|{\varpi }_3|(u_2-u_1)}{{(n+1)}^2}\sum _{i=1}^{\infty }\frac{{\varpi }_k\left(i\right){\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}^i}{k{\mathsf{\Gamma }}_k(\varrho ki+\lambda +k)}{\left(\frac{k}{\varrho pki+p\lambda +k}\right)}^{\frac{1}{p}}\hfill \\ \hfill & \times \left[{\left({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right){|}^q+{\varpi }_2{\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|}^q+{\varpi }_3\left(\frac{(2n+1)|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+{\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q}{2(n+1)}\right)\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right){|}^q+{\varpi }_2{\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|}^q+{\varpi }_3\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+(2n+1){\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q}{2(n+1)}\right)\right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & =\frac{|{\varpi }_3|(u_2-u_1)}{{(n+1)}^2}{\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_1,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]\hfill \\ \hfill & \times \left[{\left({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right){|}^q+{\varpi }_2{\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|}^q+{\varpi }_3\left(\frac{(2n+1)|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+{\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q}{2(n+1)}\right)\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right){|}^q+{\varpi }_2{\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|}^q+{\varpi }_3\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+(2n+1){\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q}{2(n+1)}\right)\right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

This completes the proof. □

3. Special Cases

In this section, we discuss some special cases of results obtained in the previous section.

Corollary 1.

Under the assumptions of Theorem 1, if we choose $\mathsf{\Psi }\left(u_1\right)=u_1$, then

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+u_2}{2}\right)\right)\hfill \\ \hfill & \le \frac{1}{2k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]}\hfill \\ & \times \left[I_{{({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2))\right.\hfill \\ \hfill & \left.+I_{{({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1))\right]\hfill \\ \hfill & \le \frac{\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}{2}\hfill \\ \hfill & \le {\varpi }_1\mathsf{\Phi }\left({\upsilon }_1\right)+{\varpi }_2\mathsf{\Phi }\left({\upsilon }_2\right)+{\varpi }_3\left[\frac{\mathsf{\Phi }\left(u_1\right)+\mathsf{\Phi }\left(u_2\right)}{2}\right].\hfill \end{array}$$

Corollary 2.

Under the assumptions of Theorem 1, if we take ${\varpi }_1=1={\varpi }_2$ and ${\varpi }_3=-1$, then

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left({\upsilon }_1+{\upsilon }_2-\left(\frac{u_1+u_2}{2}\right)\right)\hfill \\ \hfill & \le \frac{1}{2k{(u_2-u_1)}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{(-1)}^{\varrho }\right]}\left[I_{{\left[{\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_1)\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_2)\right)\right.\hfill \\ \hfill & \left.+I_{{\left[{\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_2)\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_1)\right)\right]\hfill \\ \hfill & \le \frac{\mathsf{\Phi }({\upsilon }_1+{\upsilon }_2-u_1)+\mathsf{\Phi }({\upsilon }_1+{\upsilon }_2-u_2)}{2}\hfill \\ \hfill & \le \mathsf{\Phi }\left({\upsilon }_1\right)+\mathsf{\Phi }\left({\upsilon }_2\right)-\left[\frac{\mathsf{\Phi }\left(u_1\right)+\mathsf{\Phi }\left(u_2\right)}{2}\right].\hfill \end{array}$$

Corollary 3.

Under the assumptions of Theorem 2, if we choose $\mathsf{\Psi }\left(u_1\right)=u_1$, then

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+u_2}{2}\right)\right)\hfill \\ \hfill & \le \frac{{(n+1)}^{\frac{\lambda }{k}}}{2k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}\hfill \\ & \times \left[I_{{({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2))\right.\hfill \\ & \left.+I_{{({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1))\right]\hfill \\ \hfill & \le {\varpi }_1\mathsf{\Phi }\left({\upsilon }_1\right)+{\varpi }_2\mathsf{\Phi }\left({\upsilon }_2\right)+{\varpi }_3\left[\frac{\mathsf{\Phi }\left(u_1\right)+\mathsf{\Phi }\left(u_2\right)}{2}\right].\hfill \end{array}$$

Corollary 4.

Under the assumptions of Theorem 2, if we take ${\varpi }_1=1={\varpi }_2$ and ${\varpi }_3=-1$, then

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left({\upsilon }_1+{\upsilon }_2-\frac{u_1+u_2}{2}\right)\le \frac{{(n+1)}^{\frac{\lambda }{k}}}{2k{(u_2-u_1)}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{-1}{n+1}\right)}^{\varrho }\right]}\hfill \\ & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-\frac{u_1+n{u}_2}{n+1})\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_2)\right)\right.\hfill \\ \hfill & +\left.I_{{\left[{\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-\frac{n{u}_1+u_2}{n+1})\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_1)\right)\right]\hfill \\ \hfill & \le \frac{\mathsf{\Phi }({\upsilon }_1+{\upsilon }_2-u_1)+\mathsf{\Phi }({\upsilon }_1+{\upsilon }_2-u_2)}{2}\hfill \\ \hfill & \le \mathsf{\Phi }\left({\upsilon }_1\right)+\mathsf{\Phi }\left({\upsilon }_2\right)-\left[\frac{\mathsf{\Phi }\left(u_1\right)+\mathsf{\Phi }\left(u_2\right)}{2}\right].\hfill \end{array}$$

Corollary 5.

Under the assumptions of Theorem 2, if we choose ${\varpi }_1=0={\varpi }_2$ and ${\varpi }_3=1$, then

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left(\frac{u_1+u_2}{2}\right)\le \frac{{(n+1)}^{\frac{\lambda }{k}}}{2k{(u_2-u_1)}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{1}{n+1}\right)}^{\varrho }\right]}\hfill \\ & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}\left(\frac{u_1+n{u}_2}{n+1}\right)\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}\left(u_2\right)\right)+I_{{\left[{\mathsf{\Psi }}^{-1}\left(\frac{n{u}_1+u_2}{n+1}\right)\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}\left(u_1\right)\right)\right]\hfill \\ \hfill & \le \frac{\mathsf{\Phi }\left(u_1\right)+\mathsf{\Phi }\left(u_2\right)}{2}.\hfill \end{array}$$

Corollary 6.

If we choose ${\varpi }_1=0={\varpi }_2$ and ${\varpi }_3=1$ in Corollary 3, then we obtain the Hermite-Hadamard’s inequality for generalized k-Raina’s integral

$$\begin{array}{cc}\hfill & \mathsf{\Phi }\left(\frac{u_1+u_2}{2}\right)\le \frac{{(n+1)}^{\frac{\lambda }{k}}}{2{(u_2-u_1)}^{\frac{\lambda }{k}}{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{1}{n+1}\right)}^{\varrho }\right]}\hfill \\ & \times \left[I_{{\left(\frac{u_1+n{u}_2}{n+1}\right)}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\left(u_2\right)+I_{{\left(\frac{n{u}_1+u_2}{n+1}\right)}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\left(u_1\right)\right]\le \frac{\mathsf{\Phi }\left(u_1\right)+\mathsf{\Phi }\left(u_2\right)}{2}.\hfill \end{array}$$

Corollary 7.

Under the all assumptions of Theorem 3, if we take $\mathsf{\Psi }\left(u_1\right)=u_1$, then

$$\begin{array}{cc}\hfill & |{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\frac{\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)+\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}{2}\hfill \\ & -\frac{1}{2k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\times [I_{{({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\hfill \\ & +I_{{({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\left]\right|\hfill \\ \hfill & \le {\varpi }_3(u_2-u_1){\mathcal{F}}_{\varrho ,\lambda +2k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left({\varpi }_3\right)}^{\varrho }\right]\hfill \\ & \times \left[{\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|+{\varpi }_3\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+|{\mathsf{\Phi }}^{\prime }\left(u_2\right)|}{2}\right)\right].\hfill \end{array}$$

Corollary 8.

Under the all assumptions of Theorem 3, if we choose ${\varpi }_1=0={\varpi }_2$ and ${\varpi }_3=1$, then

$$\begin{array}{cc}\hfill & |{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }\right]\left[\frac{\mathsf{\Phi }\left(u_1\right)+\mathsf{\Phi }\left(u_2\right)}{2}\right]-\frac{1}{2k{\left[(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\hfill \\ \hfill & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}\left(u_2\right)\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}\left(u_1\right)\right)+I_{[{\mathsf{\Psi }}^{-1}{\left(u_1\right)}^{]}+;\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}\left(u_2\right)\right)\right]|\hfill \\ \hfill & \le (u_2-u_1){\mathcal{F}}_{\varrho ,\lambda +2k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }\right]\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+|{\mathsf{\Phi }}^{\prime }\left(u_2\right)|}{2}\right).\hfill \end{array}$$

Corollary 9.

Under the all assumptions of Theorem 3, if we take ${\varpi }_1=1={\varpi }_2$ and ${\varpi }_3=-1$, then

$$\begin{array}{cc}\hfill & |{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }\right]\frac{\mathsf{\Phi }({\upsilon }_1+{\upsilon }_2-u_1)+\mathsf{\Phi }({\upsilon }_1+{\upsilon }_2-u_2)}{2}-\frac{1}{2k{\left[(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\hfill \\ \hfill & \times \left[I_{{\left[{\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_1)\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_2)\right)\right.\hfill \\ \hfill & +\left.I_{{\left[{\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_2)\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_1)\right)\right]|\hfill \\ \hfill & \le (u_2-u_1){\mathcal{F}}_{\varrho ,\lambda +2k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{(-1)}^{\varrho }\right]\left[|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)|-\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+|{\mathsf{\Phi }}^{\prime }\left(u_2\right)|}{2}\right)\right].\hfill \end{array}$$

Corollary 10.

Under the all assumptions of Theorem 4, if we choose $\mathsf{\Psi }\left(u_1\right)=u_1$, then

$$\begin{array}{cc}\hfill & |\frac{{(n+1)}^{\frac{\lambda }{k}-1}}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\left[I_{{({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right))}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right.\hfill \\ \hfill & +\left.I_{{({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right))}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right]\hfill \\ \hfill & -\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{n+1}\hfill \\ \hfill & \times \left[\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right)\right)+\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\right]|\hfill \\ \hfill & \le \frac{2k|{\varpi }_3|(u_2-u_1){\mathcal{F}}_{\varrho ,\lambda +2k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{{(n+1)}^2}\hfill \\ \hfill & \times \left[{\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+{\varpi }_2\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|+{\varpi }_3\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+|{\mathsf{\Phi }}^{\prime }\left(u_2\right)|}{2}\right)\right].\hfill \end{array}$$

Corollary 11.

Under the all assumptions of Theorem 4, if we take ${\varpi }_1=0={\varpi }_2$ and ${\varpi }_3=1$, then

$$\begin{array}{cc}\hfill & \left|\frac{{(n+1)}^{\frac{\lambda }{k}-1}}{k{(u_2-u_1)}^{\frac{\lambda }{k}}}\left[I_{{\left[{\mathsf{\Psi }}^{-1}\left(\frac{u_1+n{u}_2}{n+1}\right)\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}\left(u_2\right)\right)+I_{{\left[{\mathsf{\Psi }}^{-1}\left(\frac{n{u}_1+u_2}{n+1}\right)\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}\left(u_1\right)\right)\right]\right.\hfill \\ \hfill & -\left.\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{1}{n+1}\right)}^{\varrho }\right]}{n+1}\left[\mathsf{\Phi }\left(\frac{u_1+n{u}_2}{n+1}\right)+\mathsf{\Phi }\left(\frac{n{u}_1+u_2}{n+1}\right)\right]\right|\hfill \\ \hfill & \le \frac{2k(u_2-u_1){\mathcal{F}}_{\varrho ,\lambda +2k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{1}{n+1}\right)}^{\varrho }\right]}{{(n+1)}^2}\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+|{\mathsf{\Phi }}^{\prime }\left(u_2\right)|}{2}\right).\hfill \end{array}$$

Corollary 12.

Under the all assumptions of Theorem 4, if we choose ${\varpi }_1=1={\varpi }_2$ and ${\varpi }_3=-1$, then

$$\begin{array}{cc}\hfill & \left|\frac{{(n+1)}^{\frac{\lambda }{k}-1}}{k{(u_2-u_1)}^{\frac{\lambda }{k}}}\left[I_{{\left[{\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_2)\right)\right.\right.\hfill \\ \hfill & +\left.I_{{\left[{\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_1)\right)\right]\hfill \\ \hfill & -\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{-1}{n+1}\right)}^{\varrho }\right]}{n+1}\hfill \\ & \times \left[\mathsf{\Phi }\left({\upsilon }_1+{\upsilon }_2-\left(\frac{u_1+n{u}_2}{n+1}\right)\right)+\mathsf{\Phi }\left({\upsilon }_1+{\upsilon }_2-\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\right]|\hfill \\ \hfill & \le \frac{2k(u_2-u_1){\mathcal{F}}_{\varrho ,\lambda +2k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{-1}{n+1}\right)}^{\varrho }\right]}{{(n+1)}^2}\hfill \\ & \times \left[|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right)|+|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)|-\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right)|+|{\mathsf{\Phi }}^{\prime }\left(u_2\right)|}{2}\right)\right].\hfill \end{array}$$

Corollary 13.

Under the all assumptions of Theorem 5, if we take $\mathsf{\Psi }\left(u_1\right)=u_1$, then

$$\begin{array}{cc}\hfill & |\frac{{(n+1)}^{\frac{\lambda }{k}-1}}{k{\left[{\varpi }_3(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\left[I_{{[{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right)]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)\right.\hfill \\ \hfill & \left.+I_{{[{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right)]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)\right]-\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]}{n+1}\hfill \\ \hfill & \times \left[\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{u_1+n{u}_2}{n+1}\right)\right)+\mathsf{\Phi }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\right]|\hfill \\ \hfill & \le \frac{|{\varpi }_3|(u_2-u_1)}{{(n+1)}^2}{\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_1,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{{\varpi }_3}{n+1}\right)}^{\varrho }\right]\hfill \\ \hfill & \times \left[{\left({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right){|}^q+{\varpi }_2{\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|}^q+{\varpi }_3\left(\frac{(2n+1)|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+{\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q}{2(n+1)}\right)\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left({\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right){|}^q+{\varpi }_2{\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|}^q+{\varpi }_3\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+(2n+1){\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q}{2(n+1)}\right)\right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

Corollary 14.

Under the all assumptions of Theorem 5, if we choose ${\varpi }_1=0={\varpi }_2$ and ${\varpi }_3=1$, then

$$\begin{array}{cc}\hfill & |\frac{{(n+1)}^{\frac{\lambda }{k}-1}}{k{\left[(u_2-u_1)\right]}^{\frac{\lambda }{k}}}\left[I_{{\left[{\mathsf{\Psi }}^{-1}\left(\frac{u_1+n{u}_2}{n+1}\right)\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}\left(u_2\right)\right)+I_{{\left[{\mathsf{\Psi }}^{-1}\left(\frac{n{u}_1+u_2}{n+1}\right)\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}\left(u_1\right)\right)\right]\hfill \\ \hfill & -\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{1}{n+1}\right)}^{\varrho }\right]}{n+1}\left[\mathsf{\Phi }\left(\frac{u_1+n{u}_2}{n+1}\right)+\mathsf{\Phi }\left(\frac{n{u}_1+u_2}{n+1}\right)\right]|\hfill \\ \hfill & \le \frac{(u_2-u_1)}{{(n+1)}^2}{\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_1,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{1}{n+1}\right)}^{\varrho }\right]\hfill \\ \hfill & \times \left[{\left(\frac{(2n+1)|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+{\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q}{2(n+1)}\right)}^{\frac{1}{q}}{\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+(2n+1){\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q}{2(n+1)}\right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

Corollary 15.

Under the all assumptions of Theorem 5, if we take ${\varpi }_1=1={\varpi }_2$ and ${\varpi }_3=-1$, then

$$\begin{array}{cc}\hfill & |\frac{{(n+1)}^{\frac{\lambda }{k}-1}}{k{(u_2-u_1)}^{\frac{\lambda }{k}}}\left[I_{{\left[{\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-\left(\frac{u_1+n{u}_2}{n+1}\right))\right]}^{-};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_2)\right)\right.\hfill \\ \hfill & \left.+I_{{\left[{\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-\left(\frac{n{u}_1+u_2}{n+1}\right))\right]}^{+};\omega }^{\varpi ,k}\mathsf{\Phi }\circ \mathsf{\Psi }\left({\mathsf{\Psi }}^{-1}({\upsilon }_1+{\upsilon }_2-u_1)\right)\right]\hfill \\ & -\frac{{\mathcal{F}}_{\varrho ,\lambda +k}^{\varpi ,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{-1}{n+1}\right)}^{\varrho }\right]}{n+1}\hfill \\ \hfill & \times \left[\mathsf{\Phi }\left({\upsilon }_1+{\upsilon }_2-\left(\frac{u_1+n{u}_2}{n+1}\right)\right)+\mathsf{\Phi }\left({\upsilon }_1+{\upsilon }_2-\left(\frac{n{u}_1+u_2}{n+1}\right)\right)\right]|\hfill \\ \hfill & \le \frac{(u_2-u_1)}{{(n+1)}^2}{\mathcal{F}}_{\varrho ,\lambda +k}^{{\varpi }_1,k}\left[\omega {(u_2-u_1)}^{\varrho }{\left(\frac{-1}{n+1}\right)}^{\varrho }\right]\hfill \\ \hfill & \times \left[{\left(|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right){|}^q+{\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|}^q-\left(\frac{(2n+1)|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+{\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q}{2(n+1)}\right)\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left(|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_1\right){|}^q+{\left|{\mathsf{\Phi }}^{\prime }\left({\upsilon }_2\right)\right|}^q-\left(\frac{|{\mathsf{\Phi }}^{\prime }\left(u_1\right){|}^q+(2n+1){\left|{\mathsf{\Phi }}^{\prime }\left(u_2\right)\right|}^q}{2(n+1)}\right)\right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

4. Applications

In this section, we discuss some applications of our main results.

Applications to Special Means

First of all, we recall some previously known concepts regarding special means. For ${\upsilon }_1\ne {\upsilon }_2$, we have

1.

The arithmetic mean: $A({\upsilon }_1,{\upsilon }_2)=\frac{{\upsilon }_1+{\upsilon }_2}{2},$

2.

The generalized logarithmic mean: $L_n({\upsilon }_1,{\upsilon }_2)={\left[\frac{{{\upsilon }_2}^{n+1}-{{\upsilon }_1}^{n+1}}{({\upsilon }_2-{\upsilon }_1)(n+1)}\right]}^{\frac{1}{n}}$, where n belongs to $\mathbb{Z}\backslash \{-1,0\}$.

Proposition 1.

Under all the assumptions of Theorem 1, we have

$$\begin{array}{cc}\hfill & A^{\frac{r}{l}+2}(2({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2),{\varpi }_3(u_1+u_2))\hfill \\ \hfill & \le \frac{l}{r+2l}{L}_{\frac{r}{l}+2}^{\frac{r}{l}+2}[{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1,{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2]\hfill \\ \hfill & \le \frac{2l}{r+2l}A\left({{\upsilon }_1}^{\frac{r}{l}+2},{{\upsilon }_2}^{\frac{r}{l}+2}\right)+\frac{{\varpi }_{3}l}{r+2l}A\left({u_1}^{\frac{r}{l}+2},{u_2}^{\frac{r}{l}+2}\right).\hfill \end{array}$$

Proof. 

The proof is directly obtained from Theorem 1, taking $\lambda =k=1$, $i=0,$ convex function $\mathsf{\Phi }\left(x\right)=\frac{l}{r+2l}{x}^{\frac{r}{l}+2},$ and using the fact $\varpi \left(0\right)=1$. □

Proposition 2.

Under all the assumptions of Theorem 3, we have

$$\begin{array}{cc}\hfill & |\frac{l}{r+2l}A\left({({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)}^{\frac{r}{l}+2},{({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)}^{\frac{r}{l}+2}\right)\hfill \\ \hfill & -\frac{l}{r+2l}{L}_{\frac{r}{l}+2}^{\frac{r}{l}+2}[{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1,{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2]|\hfill \\ \hfill & \le \frac{|{\varpi }_3|(u_2-u_1)}{2}\left[2A\left({\varpi }_1{{\upsilon }_1}^{\frac{r}{l}+1},{\varpi }_2{{\upsilon }_2}^{\frac{r}{l}+1}\right)+{\varpi }_{3}A\left({u_1}^{\frac{r}{l}+1},{u_2}^{\frac{r}{l}+1}\right)\right].\hfill \end{array}$$

Proof. 

The proof is directly obtained from Theorem 3, choosing $\lambda =k=1$, $i=0,$ convex function $\mathsf{\Phi }\left(x\right)=\frac{l}{r+2l}{x}^{\frac{r}{l}+2},$ and using the fact $\varpi \left(0\right)=1$. □

Proposition 3.

Under the assumptions of Theorem 4, we have

$$\begin{array}{cc}\hfill & |A^{\frac{r}{l}+2}\left(2({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2)+{\varpi }_3(u_1+u_2)\right)\hfill \\ & -L_{\frac{r}{l}+2}^{\frac{r}{l}+2}[{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1,{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2]|\hfill \\ \hfill & \le \frac{|{\varpi }_3|(u_2-u_1)}{8}\left[2A\left({\varpi }_1{{\upsilon }_1}^{\frac{r}{l}+1},{\varpi }_2{{\upsilon }_2}^{\frac{r}{l}+1}\right)+{\varpi }_{3}A\left({u_1}^{\frac{r}{l}+1},{u_2}^{\frac{r}{l}+1}\right)\right].\hfill \end{array}$$

Proof. 

The proof is directly obtained from Theorem 4, taking $\lambda =k=1$, $i=0,$ convex function $\mathsf{\Phi }\left(x\right)=\frac{l}{r+2l}{x}^{\frac{r}{l}+2},$ and using the fact $\varpi \left(0\right)=1$. □

Proposition 4.

Under the assumptions of Theorem 5, we have

$$\begin{array}{cc}\hfill & |A^{\frac{r}{q}+2}\left(2({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2)+{\varpi }_3(u_1+u_2)\right)\hfill \\ & -\frac{q}{r+2q}{L}_{\frac{r}{q}+2}^{\frac{r}{q}+2}[{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1,{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2]|\hfill \\ \hfill & \le \frac{|{\varpi }_3|(u_2-u_1)}{4}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\left[{\left(2A\left({\varpi }_1{{\upsilon }_1}^{(\frac{r}{l}+1)q},{\varpi }_2{{\upsilon }_2}^{(\frac{r}{l}+1)q}\right)+\frac{{\varpi }_3}{2}A\left(3{u_1}^{(\frac{r}{l}+1)q},{u_2}^{(\frac{r}{l}+1)q}\right)\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & \left.+{\left(2A\left({\varpi }_1{{\upsilon }_1}^{(\frac{r}{l}+1)q},{\varpi }_2{{\upsilon }_2}^{(\frac{r}{l}+1)q}\right)+\frac{{\varpi }_3}{2}A\left({u_1}^{(\frac{r}{l}+1)q},3{u_2}^{(\frac{r}{l}+1)q}\right)\right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

Proof. 

The proof is directly obtained from Theorem 5, choosing $\lambda =k=1$, $i=0,$ convex function $\mathsf{\Phi }\left(x\right)=\frac{l}{r+2l}{x}^{\frac{r}{l}+2},$ and using the fact $\varpi \left(0\right)=1$. □

Here, we present an error bound in connection with Theorem 3. For this, we fix ${\varpi }_1,{\varpi }_2\in [0,1]$ and ${\varpi }_3\in (0,1]$ with ${\varpi }_1+{\varpi }_2+{\varpi }_3=1$. For ${\upsilon }_2>{\upsilon }_1>0$, let $\widehat{S}:{\upsilon }_1={u_1}_0<{u_1}_1<{u_1}_2<\dots <{u_1}_i<{u_1}_{i+1}<\dots <{u_1}_n={\upsilon }_2$ be a partition of $[{\upsilon }_1,{\upsilon }_2]$ and ${u_1}_{i1},{u_1}_{i2}\in [{u_1}_i,{u_1}_{i+1}].$

$$\begin{array}{cc}\hfill & T(\widehat{S},\mathsf{\Phi }):={\varpi }_3\sum _{i=0}^n\left[\frac{\mathsf{\Phi }({\varpi }_1{u_1}_i+{\varpi }_2{u_1}_{i+1}+{\varpi }_3{u_1}_{i1})+\mathsf{\Phi }({\varpi }_1{u_1}_i+{\varpi }_2{u_1}_{i+1}+{\varpi }_3{u_1}_{i2})}{2}\right]h_i,\hfill \\ \hfill & {\int }_{{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1}^{{\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2}\mathsf{\Phi }\left(u_1\right)\mathrm{d}{u}_1:=T(\widehat{S},\mathsf{\Phi })+\overline{R}(\widehat{S},\mathsf{\Phi }),\hfill \end{array}$$

where $T(\widehat{S},\mathsf{\Phi })$ is the trapezium quadrature formula and $\overline{R}(\widehat{S},\mathsf{\Phi })$ is the remainder term (error estimation).

Proposition 5.

Under all the assumptions of Theorem 3, then

$$\begin{array}{c}\hfill |\overline{R}(\widehat{S},\mathsf{\Phi })|\le \frac{{{\varpi }_3}^2}{2}\sum _{i=0}^n{h}_i^2\left[{\varpi }_1|{\mathsf{\Phi }}^{\prime }\left({u_1}_i\right)|+{\varpi }_2\left|{\mathsf{\Phi }}^{\prime }\left({u_1}_{i+1}\right)\right|+{\varpi }_3\frac{|{\mathsf{\Phi }}^{\prime }\left({u_1}_{i1}\right)|+|{\mathsf{\Phi }}^{\prime }\left({u_1}_{i2}\right)|}{2}\right],\end{array}$$

where $h_i:={u_1}_{i+1}-{u_1}_i$.

Proof. 

Using Theorem 3 over subinterval $[{u_1}_i,{u_1}_{i+1}]$ of closed interval $[{\upsilon }_1,{\upsilon }_2]$, taking sum with respect to index i from 0 to $n-1$, and choosing $\lambda =k=1$, we get our required result. □

Remark 3.

Using the same technique as in above results, we can derive many new error estimations using our main results. We omit here their proofs and the details are left to the interested reader.

Now we provide some applications to q-digamma function [20]. Suppose $0<q<1$, then q-digamma function ${\chi }_q\left(u\right)$ is given as:

$$\begin{array}{cc}\hfill {\chi }_q\left(u\right)& =-ln(1-q)+ln\left(q\right)\sum _{i=0}^{\infty }\frac{q^{i+u}}{1-q^{i+u}}\hfill \\ \hfill & =-ln(1-q)+ln\left(q\right)\sum _{i=0}^{\infty }\frac{q^{iu}}{1-q^{iu}}.\hfill \end{array}$$

For $q>1$ and $u>0,$ then q-digamma function ${\chi }_q\left(u\right)$ can be given as:

$$\begin{array}{cc}\hfill {\chi }_q\left(u\right)& =-ln(q-1)+ln\left(q\right)\left[u-\frac{1}{2}-\sum _{i=0}^{\infty }\frac{q^{-(i+u)}}{1-q^{-(i+u)}}\right]\hfill \\ \hfill & =-ln(q-1)+ln\left(q\right)\left[u-\frac{1}{2}-\sum _{i=0}^{\infty }\frac{q^{-iu}}{1-q^{-iu}}\right].\hfill \end{array}$$

From the above definition, it is clear that ${\chi }_q^{\prime }\left(x\right)$ is completely monotone function on $(0,\infty )$ for $q>0$. This implies that ${\chi }_q^{\prime }\left(x\right)$ is convex function. Using this fact, we can establish the following new important results regarding q-digamma function.

Proposition 6.

Under the assumptions of Theorem 1, we have

$$\begin{array}{cc}\hfill & {\chi }_q^{\prime }\left({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3\frac{u_1+u_2}{2}\right)\hfill \\ & \le \frac{{\chi }_q({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)-{\chi }_q({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)}{u_2-u_1}\hfill \\ \hfill & \le {\varpi }_1{\chi }_q^{\prime }\left({\upsilon }_1\right)+{\varpi }_2{\chi }_q^{\prime }\left({\upsilon }_2\right)+{\varpi }_3\frac{{\chi }_q^{\prime }\left(u_1\right)+{\chi }_q^{\prime }\left(u_2\right)}{2}.\hfill \end{array}$$

Proof. 

The proof directly follows from Theorem 1, taking $\lambda =k=1$, $i=0,\mathsf{\Phi }\left(x\right)={\chi }_q^{\prime }\left(x\right)$ and using the fact $\varpi \left(0\right)=1$. □

Proposition 7.

Under the assumptions of Theorem 3, we have

$$\begin{array}{cc}\hfill & |\frac{{\chi }_q^{\prime }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)+{\chi }_q^{\prime }({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)}{2}\hfill \\ & -\frac{{\chi }_q({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_2)-{\chi }_q({\varpi }_1{\upsilon }_1+{\varpi }_2{\upsilon }_2+{\varpi }_3{u}_1)}{u_2-u_1}|\hfill \\ \hfill & \le \frac{|{\varpi }_3|(u_2-u_1)}{2}\left[{\varpi }_1{{\chi }^{″}}_q\left({\upsilon }_1\right)+{\varpi }_2{{\chi }^{″}}_q\left({\upsilon }_2\right)+{\varpi }_3\frac{{{\chi }^{″}}_q\left(u_1\right)+{{\chi }^{″}}_q\left(u_2\right)}{2}\right].\hfill \end{array}$$

Proof. 

The proof directly follows from Theorem 3, choosing $\lambda =k=1$, $i=0,\mathsf{\Phi }\left(x\right)={\chi }_q^{\prime }\left(x\right)$ and using the fact $\varpi \left(0\right)=1$. □

Remark 4.

Using the same technique as in above proved relations, we can obtain several interesting inequalities pertaining to q-digamma function applying our main results. We omit here their proofs.

5. Graphical Analysis

In this section, we discuss the pictorial illustrations of our main outcome, which are helpful to understand theoretical results.

If we choose $\mathsf{\Phi }\left(x\right)=x^2$ with ${\upsilon }_1=0,{\upsilon }_2=3,u_1=1$ and $u_2=2$ in Theorem 3 (Figure 1), then we have

$$\begin{array}{cc}\hfill & \left|\frac{5}{2\lambda \mathsf{\Gamma }\left(\lambda \right)}-\frac{1}{2\mathsf{\Gamma }\left(\lambda \right)}\left[\frac{5}{\lambda }-\frac{2}{1+\lambda }+\frac{2}{2+\lambda }\right]\right|\le \frac{3(2^{\lambda }-1)}{2^{\lambda }\mathsf{\Gamma }(\lambda +2)}.\hfill \end{array}$$

If we choose $\mathsf{\Phi }\left(x\right)=x^2$ with ${\upsilon }_1=0,{\upsilon }_2=3,u_1=1$ and $u_2=2$ in Theorem 4 (Figure 2), then we have

$$\begin{array}{c}\hfill \left|\frac{18{\lambda }^2+54\lambda +40}{8\lambda \gamma \left(\lambda \right)({\lambda }^2+3\lambda +2)}-\frac{9}{4\mathsf{\Gamma }(\lambda +1)}\right|\le \frac{3}{2\mathsf{\Gamma }(\lambda +2)}.\end{array}$$

If we choose $\mathsf{\Phi }\left(x\right)=x^2$ with ${\upsilon }_1=0,{\upsilon }_2=3,u_1=1$ and $u_2=2$ in Theorem 5 (Figure 3), then we have

$$\begin{array}{c}\hfill \left|\frac{18{\lambda }^2+54\lambda +40}{8\lambda \gamma \left(\lambda \right)({\lambda }^2+3\lambda +2)}-\frac{9}{4\mathsf{\Gamma }(\lambda +1)}\right|\le \frac{1}{4\mathsf{\Gamma }(\lambda +1)}{\left(\frac{1}{2\mathsf{\Gamma }\left(\lambda \right)+3}\right)}^{\frac{1}{2}}\left(\sqrt{29}+\sqrt{26}\right).\end{array}$$

6. Conclusions

In this paper, we have derived some new generalized fractional analogues of Mercer type inequalities applying the convexity property of the functions and Raina’s function. We also discussed several new special cases which showed that our results are quite unifying. The efficiency of our results was demonstrated with several interesting applications from our results pertaining to special means, error bounds and q-digamma functions. To the best of our knowledge, these results are new in the literature. Since the class of convex functions have large applications in many mathematical areas, they can be applied to obtain several results in convex analysis, special functions, quantum mechanics, related optimization theory, and mathematical inequalities. They may stimulate further research in different areas of pure and applied sciences. Using the techniques and ideas of this paper may inspire and motivate further research in this energetic field.