On Riemann–Liouville Integral via Strongly Modified (h,m)-Convex Functions

Abstract

The generalization of strongly convex and strongly m-convex functions is presented in this paper. We began by proving the properties of a strongly modified $(h,m)$-convex function. The Schur inequality and the Hermite–Hadamard (H-H) inequalities are proved for the proposed class. Moreover, H-H inequalities are also proved in the context of Riemann–Liouville (R-L) integrals. Some examples and graphs are also presented in order to show the existence of this newly defined class.

1. Introduction

A significant discipline in mathematics, applicable in many fields, is convex analysis [1]. It is utilized in the investigation of convex function behavior and applications across several domains [2]. The examination of mathematical inequalities and the solution of optimization problems depend substantially on convex functions (CFs) [3,4]. These functions have some well-known inequalities as direct implications. Because of the exceptional properties of these functions, they are very crucial in multiple fields including economics [5], geometry [6] and mathematical analysis [4].

Inequalities are very helpful in determining the behavior of a function over an interval. They are also valuable in finding the maximum and minimum values of the function. Many areas of mathematics, including fractional calculus, discrete fractional calculus, and mathematical analysis, can benefit from the use of integral inequalities on various kinds of convex functions (see references [7,8,9]).

In the field of convex analysis, one of the famous inequalities is the Hermite–Hadamard inequality. The studies of Hermite that were published in Mathesis 3 in 1883 and by Hadamard ten years later are where it all began. The Hermite–Hadamard inequality has been widely used for a long time after being known as Hadamard’s inequality [10,11]. For a convex function $\xi :B_1\subset \mathbb{R}\to \mathbb{R}$, the H-H inequality [12] is given as

$$\xi \left({\displaystyle \frac{{\varphi }_1+{\varphi }_2}{2}}\right)\le {\displaystyle \frac{1}{{\varphi }_2-{\varphi }_1}}{\int }_{{\varphi }_1}^{{\varphi }_2}\xi \left(k\right)dk\le {\displaystyle \frac{\xi \left({\varphi }_1\right)+\xi \left({\varphi }_2\right)}{2}},forany{\varphi }_1,{\varphi }_2\in B_1.$$

The H-H inequalities are utilized in many different domains. For example, in the subject of optimization theory, they are used to determine the limits of functions over given intervals [13]. In the domain of signal processing, H-H inequalities are also useful for identifying the characteristics of signals and creating efficient filters that guarantee the smooth processing of signals [13].

The use of derivatives and integrals under fractional orders (i.e., orders that are not natural numbers or integers) is the main emphasis of the branch of mathematics known as fractional calculus [14,15]. A number of scholars have expressed interest in the theory of fractional calculus, which has grown quickly (see [8,16,17,18,19,20]).

Inspired by the work of Angulo et. al. on strongly h-CFs in [21], Lara et. al. on strongly m-CFs in [22], and Toader on m-CFs and modified h-CFs in [23], a new class is introduced which generalizes these existing classes. Moreover, our main results generalize the results along with the applications presented in [21,22,23].

The paper is organized in the following sequence. The definitions of some types of convex functions are revised in Section 2. The basic properties of strongly modified $(h,m)$-convex functions are demonstrated in Section 3. The Schur inequality for the newly defined class of convexity is established in Section 4. Section 5 presents the H-H inequalities for the introduced class of convex functions. Section 6 provides an overview of the whole work.

2. Preliminaries

The following information is deployed throughout the paper:

(a)

Suppose $h:[0,1]\to {\mathbb{R}}^{+}$ is a non-zero, non-negative and super multiplicative function;

(b)

Suppose $u\in [0,1]$, $l_1>0$, and $m\in [0,1]$.

A function $\xi :B_1\subset \mathbb{R}\to \mathbb{R}$ is a modified h-CF [24] if

$$\xi (u{\varphi }_1+(1-u){\varphi }_2)\le h\left(u\right)\xi \left({\varphi }_1\right)+(1-h\left(u\right))\xi \left({\varphi }_2\right)$$

holds, $\forall {\varphi }_1,{\varphi }_2\in B_1$.

A real valued function $\xi :[0,b]\to \mathbb{R}$ with $b>0$ is an m-CF [23] if

$$\xi (u{\phi }_1+m(1-u){\phi }_2)\le u\xi \left({\phi }_1\right)+m(1-u)\xi \left({\phi }_2\right)$$

holds, $\forall {\varphi }_1,{\varphi }_2\in [0,b]$.

A function $\xi :B_1\subset \mathbb{R}\to \mathbb{R}$ is a strongly convex function [25] if

$$\xi (u{\varphi }_1+(1-u){\varphi }_2)\le u\xi \left({\varphi }_1\right)+(1-u)\xi \left({\varphi }_2\right)-l_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2$$

holds, $\forall {\varphi }_1,{\varphi }_2\in B_1$.

A function $\xi :[0,b]\to \mathbb{R}$ is a strongly m-CF [22] if

$$\xi (u{\varphi }_1+m(1-u){\varphi }_2)\le u\xi \left({\varphi }_1\right)+m(1-u)\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2$$

holds, $\forall {\varphi }_1,{\varphi }_2\in [0,b]$.

A function $\xi$: $B_1\subset \mathbb{R}\to \mathbb{R}$ is called super multiplicative [26] if

$$\xi \left({\varphi }_1{\varphi }_2\right)\ge \xi \left({\varphi }_1\right)\xi \left({\varphi }_2\right),\forall {\varphi }_1,{\varphi }_2\in B_1.$$

Suppose $\xi \in L_1[c,d]$, the R-L integrals $M_{c^{+}}^{\beta }\xi$ and $M_{d^{-}}^{\beta }\xi$ of order $\gamma >0$ are defined as [27]

$$M_{c^{+}}^{\gamma }\xi \left(\varphi \right)={\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}\underset{c}{\overset{\varphi }{\int }}{(\varphi -k)}^{\gamma -1}\xi \left(k\right)dk,with\varphi >c.$$

$$M_{d^{-}}^{\gamma }\xi \left(\varphi \right)={\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}\underset{\varphi }{\overset{d}{\int }}{(k-\varphi )}^{\gamma -1}\xi \left(k\right)dk,with\varphi <d.$$

where $\Gamma (.)$ is a classical Euler Gamma function [28].

3. Main Results

The section introduces a strongly modified $(h,m)$-convex function (SM $(h,m)$-CF).

A function $\xi :[0,b]\to \mathbb{R}$ with $b>0$ is a strongly modified $(h,m)$-convex function with modulus $l_1$ if

$$\xi (u{\varphi }_1+m(1-u){\varphi }_2)\le h\left(u\right)\xi \left({\varphi }_1\right)+m(1-h\left(u\right))\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2$$

(1)

holds $\forall {\varphi }_1,{\varphi }_2\in [0,b]$, $l_1>0$, $m\in [0,1]$, and $u\in [0,1]$.

Remark 1. 

(a) 

By assuming $l_1=0$ and $m=1$ in (1), one obtains the modified h-CF (see [29]);

(b) 

By choosing $l_1=0$ and $h\left(u\right)=u$ in (1), one obtains the m-CF (see [23]);

(c) 

By putting $h\left(u\right)=u$ in (1), one obtains a strongly m-CF (see [22]);

(d) 

By assuming $h\left(u\right)=u$ and $m=1$ in (1), one obtains the strongly convex function (see [30]).

Note: The following information is deployed throughout the graphs of the next examples:

(a) 

The values of ${\varphi }_1$ are provided along the x-axis;

(b) 

The values of ${\varphi }_2$ are given along the y-axis;

(c) 

The values of functions on the left-hand and the right-hand sides of inequalities (2) and (3) are specified along the z-axis.

Example 1. 

For ${\varphi }_1,{\varphi }_2\in [1,\infty ),$ with ${\varphi }_1<{\varphi }_2,$ $m\in [0,1]$, $u\in [0,1],$ $l_1>0$ and $h\left(u\right)=u^2,$ the function $\xi \left(u\right)=u^2$ is SM $(h,m)$-CF.

Assume ${\varphi }_1,{\varphi }_2\in [1,\infty )$ with ${\varphi }_1<{\varphi }_2,$ $u=1/2$ $m=1/2$ and $l_1=1,$ in inequality (1), to obtain

$${\left({\displaystyle \frac{2{\varphi }_1+{\varphi }_2}{4}}\right)}^2\le {\displaystyle \frac{{\varphi }_1^2}{4}}+{\displaystyle \frac{3{\varphi }_2^2}{8}}-{\displaystyle \frac{1}{8}}{({\varphi }_1-{\varphi }_2)}^2.$$

(2)

The validity of inequality (2) is presented in Figure 1. Hence, it is SM $(h,m)$-CF.

Example 2. 

For ${\varphi }_1,{\varphi }_2\in [1,\infty ),$ $m\in [0,1],$ $u\in [0,1],$ $l_1>0$ and $h\left(u\right)=u^2,$ the function $\xi \left(u\right)={(1-u)}^2$ is SM $(h,m)$-CF.

Choose ${\varphi }_1,{\varphi }_2\in [1,\infty )$ with ${\varphi }_1<{\varphi }_2,$ $u=1/2$ and $l_1=1/2,$ in inequality (1), to obtain

$${\left(1-\left({\displaystyle \frac{2{\varphi }_1+{\varphi }_2}{4}}\right)\right)}^2\le {\displaystyle \frac{{(1-{\varphi }_1)}^2}{4}}+{\displaystyle \frac{3{(1-{\varphi }_2)}^2}{8}}-{\displaystyle \frac{{({\varphi }_1-{\varphi }_2)}^2}{16}}$$

(3)

Figure 2 presents the validity of inequality (3). Thus, it is also an SM $(h,m)$-CF.

The properties of SM $(h,m)$-CFs are established in the following lemmas.

Lemma 1. 

Suppose ξ and Φ are SM $(h,m)$-CFs, then their sum is also a SM $(h,m)$-CF.

Proof. 

For ${\varphi }_1,{\varphi }_2\in [0,b]$ with $b>0$, we have

$$\begin{array}{c}\hfill (\xi +\Phi )(u{\varphi }_1+m(1-u){\varphi }_2)=\xi (u{\varphi }_1+m(1-u){\varphi }_2)+\Phi (u{\varphi }_1+m(1-u){\varphi }_2).\end{array}$$

Since $\xi$ and $\Phi$ are SM $(h,m)$-convex functions,

$$\begin{array}{cc}\hfill (\xi +\Phi )(u{\varphi }_1+m(1-u){\varphi }_2)& \le h\left(u\right)\xi \left({\varphi }_1\right)+m(1-h\left(u\right))\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2\hfill \\ \hfill & +h\left(u\right)\Phi \left({\varphi }_1\right)+m(1-h\left(u\right))\Phi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2\hfill \\ \hfill & \le h\left(u\right)(\xi +\Phi )\left({\varphi }_1\right)+m(1-h\left(u\right))(\xi +\Phi )\left({\varphi }_2\right))\hfill \\ \hfill & -m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2.\hfill \end{array}$$

 □

Lemma 2. 

Assume that ξ is a SM $(h,m)$-CF, then for scalar $n>0$, $n\xi$ is also a SM $(h,m)$-CF.

Proof. 

Since $\xi$ is a SM $(h,m)$-convex function, therefore, for ${\varphi }_1,{\varphi }_2\in [0,b]$ with $b>0$, we have

$$\begin{array}{cc}\hfill n\xi (u{\varphi }_1+m(1-u){\varphi }_2)& \le n\left(h\left(u\right)\xi \left({\varphi }_1\right)+m(1-h\left(u\right))\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2\right)\hfill \\ \hfill & =h\left(u\right)n\xi \left({\varphi }_1\right)+m(1-h\left(u\right))n\xi \left({\varphi }_2\right)-nm{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2.\hfill \end{array}$$

 □

Lemma 3. 

Suppose $h_1,h_2$ are non-zero functions on $[0,1]$ such that $h_2\left(r\right)\le h_1\left(r\right)$. If ξ is a SM $(h_2,m)$-CF, then ξ is also a SM $(h_1,m)$-CF.

Proof. 

Since $\xi$ is a SM $(h_2,m)$-convex function, so for ${\varphi }_1,{\varphi }_2\in [0,b]$ with $b>0$, we have

$$\begin{array}{cc}\hfill \xi (u{\varphi }_1+m(1-u){\varphi }_2)& \le h_2\left(u\right)\xi \left({\varphi }_1\right)+m(1-h_2\left(u\right))\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2\hfill \\ \hfill & \le h_1\left(u\right)\xi \left({\varphi }_1\right)+m(1-h_1\left(u\right))\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2.\hfill \end{array}$$

 □

Lemma 4. 

Suppose ${\xi }_s:[0,b]\to \mathbb{R}$ are SM $(h,m)$-CFs for $s\in \mathbb{N}$ and ${\displaystyle \sum _{s=1}^d}{n}_s=1$; then, their linear combination

$$W\left(t\right)={\displaystyle \sum _{s=1}^d}{n}_s{\xi }_s\left(t\right),$$

(4)

∀$t\in [0,b]$ is also SM $(h,m)$-CF.

Proof. 

By choosing ${\varphi }_1,{\varphi }_2\in [0,b]$ with $b>0$, and $t=(u{\varphi }_1+m(1-u){\varphi }_2),$ in (4) it becomes

$$\begin{array}{c}\hfill W(u{\varphi }_1+m(1-u){\varphi }_2)=\sum _{s=1}^d{n}_s({\xi }_s(u{\varphi }_1+m(1-u){\varphi }_2).\end{array}$$

Since ${\xi }_s$ is a SM $(h,m)$-convex function, therefore

$$\begin{array}{cc}\hfill W(u{\varphi }_1+m(1-u){\varphi }_2)& \le h\left(u\right)\left(\sum _{s=1}^d{n}_s\left({\xi }_s\left({\varphi }_1\right)\right)\right)+m(1-h\left(u\right))\left(\sum _{s=1}^d{n}_s({\xi }_s\left({\varphi }_2\right))\right)\hfill \\ \hfill & -\sum _{s=1}^d{n}_s{l}_{1}mu(1-u){({\varphi }_1-{\varphi }_2)}^2\hfill \\ \hfill & =h\left(u\right)W\left({\varphi }_1\right)+m(1-h\left(u\right))W\left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2.\hfill \end{array}$$

 □

Lemma 5. 

Let ${\xi }_s:[0,b]\to \mathbb{R}$, such that $s\in \mathbb{N}$ and $b>0$, be a non-empty collection of SM $(h,m)$-CFs such that ∀ $l\in [0,b]$, ${sup}_{s\in \mathbb{N}}{\xi }_s\left(l\right)$ exists in $\mathbb{R}$, then function ξ, defined by $\xi \left(l\right)={sup}_{s\in \mathbb{N}}{\xi }_s\left(l\right)$ ∀ $l\in [0,b]$ is also SM $(h,m)$-CF.

Proof. 

By choosing ${\varphi }_1,{\varphi }_2\in [0,b]$ with $b>0$, and $l=(u{\varphi }_1+m(1-u){\varphi }_2),$ in

$$\xi \left(l\right)=\underset{s\in \mathbb{N}}{sup}{\xi }_s\left(l\right).$$

we obtain

$$\begin{array}{c}\hfill \xi (u{\varphi }_1+m(1-u){\varphi }_2)=\underset{s\in \mathbb{N}}{sup}{\xi }_s(u{\varphi }_1+m(1-u){\varphi }_2).\end{array}$$

Since ${\xi }_s$ is a SM $(h,m)$-convex function, so

$$\begin{array}{cc}\hfill \xi (u{\varphi }_1+m(1-u){\varphi }_2)& \le h\left(u\right)\left(\underset{s\in \mathbb{N}}{sup}\xi \left({\varphi }_1\right)\right)+m(1-h\left(u\right))\left(\underset{s\in \mathbb{N}}{sup}\xi \left({\varphi }_2\right)\right)\hfill \\ \hfill & -m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2\hfill \\ \hfill & =h\left(u\right)\xi \left({\varphi }_1\right)+m(1-h\left(u\right))\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2.\hfill \end{array}$$

 □

Lemma 6. 

Let ξ be a SM $(h,m)$-CF and Φ be a linear function, then $\xi \circ \Phi$ is also SM $(h,m)$-convex.

Proof. 

For ${\varphi }_1,{\varphi }_2\in [0,b]$ with $b>0$, we have

$$\begin{array}{c}\hfill (\xi \circ \Phi )(u{\varphi }_1+m(1-u){\varphi }_2)=\xi (u\Phi \left({\varphi }_1\right)+m(1-u)\Phi \left({\varphi }_2\right)).\end{array}$$

Since $\xi$ is a SM $(h,m)$-CF, therefore

$$\begin{array}{cc}\hfill (\xi \circ \Phi )(u{\varphi }_1+m(1-u){\varphi }_2)& \le h\left(u\right)\xi (\Phi \left({\varphi }_1\right))+m(1-h\left(u\right))\xi (\Phi \left({\varphi }_2\right))\hfill \\ \hfill & -m{l}_{1}u(1-u){(\Phi \left({\varphi }_1\right)-\Phi \left({\varphi }_2\right))}^2\hfill \\ \hfill & =h\left(u\right)(\xi \circ \Phi \left({\varphi }_1\right))+m(1-h\left(u\right))(\xi \circ \Phi \left({\varphi }_2\right))\hfill \\ \hfill & -m{l}_{1}u(1-u){(\Phi \left({\varphi }_1\right)-\Phi \left({\varphi }_2\right))}^2.\hfill \end{array}$$

 □

4. Schur Inequality

The following theorem provides the Schur inequality for the SM $(h,m)$-CF.

Theorem 1. 

Suppose ξ is SM $(h,m)$-CF. Then, for ${\varphi }_1,{\varphi }_2,{\varphi }_3\in [0,b]$ such that ${\varphi }_1<{\varphi }_2<{\varphi }_3,$ ${\varphi }_3-{\varphi }_1,{\varphi }_3-{\varphi }_2,{\varphi }_2-{\varphi }_1\in [0,b]$ and $m,u\in [0,1]$, we have

$$\begin{array}{ccc}& & \xi \left(\left({\displaystyle \frac{({\varphi }_3-{\varphi }_2)}{({\varphi }_3-{\varphi }_1)}}\right){\varphi }_1+m\left({\displaystyle \frac{({\varphi }_2-{\varphi }_1)}{({\varphi }_3-{\varphi }_1)}}\right){\varphi }_3\right)\times h({\varphi }_3-{\varphi }_1)\hfill \\ & & \le h({\varphi }_3-{\varphi }_2)\xi \left({\varphi }_1\right)+m\left(h({\varphi }_3-{\varphi }_1)-h({\varphi }_3-{\varphi }_2)\right)\xi \left({\varphi }_3\right)\hfill \\ & & -h({\varphi }_3-{\varphi }_1)l_{1}m({\varphi }_3-{\varphi }_2)({\varphi }_2-{\varphi }_1).\hfill \end{array}$$

(5)

Proof. 

Let ${\varphi }_1,{\varphi }_2,{\varphi }_3\in [0,b]$ be such that ${\varphi }_1<{\varphi }_2<{\varphi }_3,$ then ${\displaystyle \left(\frac{({\varphi }_3-{\varphi }_2)}{({\varphi }_3-{\varphi }_1)}\right)\in (0,1)}$ and ${\displaystyle \left(\frac{({\varphi }_2-{\varphi }_1)}{({\varphi }_3-{\varphi }_1)}\right)\in (0,1),}$ then we have

$$h({\varphi }_3-{\varphi }_2)=h\left({\displaystyle \frac{({\varphi }_3-{\varphi }_2)}{({\varphi }_3-{\varphi }_1)}}\times ({\varphi }_3-{\varphi }_1)\right)\le h\left({\displaystyle \frac{({\varphi }_3-{\varphi }_2)}{({\varphi }_3-{\varphi }_1)}}\right)\times h({\varphi }_3-{\varphi }_1).$$

Suppose $h({\varphi }_3-{\varphi }_2)>0$. Since $\xi$ is SM (h,m)-convex, so

$$\begin{array}{c}\hfill \begin{array}{cc}& \xi (us+m(1-u\left)w\right)\le h\left(u\right)\xi \left(s\right)+m(1-h\left(u\right))\xi \left(w\right)-m{l}_{1}u(1-{\varphi }_1){(s-w)}^2.\hfill \end{array}\end{array}$$

(6)

By choosing ${\displaystyle \frac{({\varphi }_3-{\varphi }_2)}{({\varphi }_3-{\varphi }_1)}=u}$, $s={\varphi }_1$ and $w={\varphi }_3$ in (6), we obtain

$$\begin{array}{ccc}& & \xi \left(\left({\displaystyle \frac{({\varphi }_3-{\varphi }_2)}{({\varphi }_3-{\varphi }_1)}}\right){\varphi }_1+m\left({\displaystyle \frac{({\varphi }_2-{\varphi }_1)}{({\varphi }_3-{\varphi }_1)}}\right){\varphi }_3\right)\le h\left({\displaystyle \frac{\left({\varphi }_3-{\varphi }_2\right)}{\left({\varphi }_3-{\varphi }_1\right)}}\right)\xi \left({\varphi }_1\right)\hfill \\ & & +m\left(1-h\left({\displaystyle \frac{\left({\varphi }_3-{\varphi }_2\right)}{\left({\varphi }_3-{\varphi }_1\right)}}\right)\right)\xi \left({\varphi }_3\right)-m{l}_1{\displaystyle \frac{({\varphi }_3-{\varphi }_2)({\varphi }_2-{\varphi }_1)}{{({\varphi }_1-{\varphi }_3)}^2}}{({\varphi }_1-{\varphi }_3)}^2.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill ⟹& & \xi \left(\left({\displaystyle \frac{({\varphi }_3-{\varphi }_2)}{({\varphi }_3-{\varphi }_1)}}\right){\varphi }_1+m\left({\displaystyle \frac{({\varphi }_2-{\varphi }_1)}{({\varphi }_3-{\varphi }_1)}}\right){\varphi }_3\right)\times h({\varphi }_3-{\varphi }_1)\hfill \\ & & \le h({\varphi }_3-{\varphi }_2)\xi \left({\varphi }_1\right)+m\left(h({\varphi }_3-{\varphi }_1)-h({\varphi }_3-{\varphi }_2)\right)\xi \left({\varphi }_3\right)\hfill \\ & & -h({\varphi }_3-{\varphi }_1)\times m{l}_1({\varphi }_3-{\varphi }_2)({\varphi }_2-{\varphi }_1).\hfill \end{array}$$

Conversely,

$$\begin{array}{ccc}& & \xi \left(\left({\displaystyle \frac{({\varphi }_3-{\varphi }_2)}{({\varphi }_3-{\varphi }_1)}}\right){\varphi }_1+m\left({\displaystyle \frac{({\varphi }_2-{\varphi }_1)}{({\varphi }_3-{\varphi }_1)}}\right){\varphi }_3\right)\times h({\varphi }_3-{\varphi }_1)\hfill \\ & & \le h({\varphi }_3-{\varphi }_2)\xi \left({\varphi }_1\right)+m\left(h({\varphi }_3-{\varphi }_1)-h({\varphi }_3-{\varphi }_2)\right)\xi \left({\varphi }_3\right)\hfill \\ & & -h({\varphi }_3-{\varphi }_1)\times m{l}_1({\varphi }_3-{\varphi }_2)({\varphi }_2-{\varphi }_1).\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & & \xi \left(\left({\displaystyle \frac{({\varphi }_3-{\varphi }_2)}{({\varphi }_3-{\varphi }_1)}}\right){\varphi }_1+m\left({\displaystyle \frac{({\varphi }_2-{\varphi }_1)}{({\varphi }_3-{\varphi }_1)}}\right){\varphi }_3\right)\le h\left({\displaystyle \frac{\left({\varphi }_3-{\varphi }_2\right)}{\left({\varphi }_3-{\varphi }_1\right)}}\right)\xi \left({\varphi }_1\right)\hfill \\ & & +m\left(1-h\left({\displaystyle \frac{\left({\varphi }_3-{\varphi }_2\right)}{\left({\varphi }_3-{\varphi }_1\right)}}\right)\right)\xi \left({\varphi }_3\right)-m{l}_1{\displaystyle \frac{({\varphi }_3-{\varphi }_2)({\varphi }_2-{\varphi }_1)}{{({\varphi }_1-{\varphi }_3)}^2}}{({\varphi }_1-{\varphi }_3)}^2.\hfill \end{array}$$

(7)

By choosing, ${\displaystyle \frac{({\varphi }_3-{\varphi }_2)}{({\varphi }_3-{\varphi }_1)}=u}$, $s={\varphi }_1$ and $w={\varphi }_3$ in (7), one obtains

$$\begin{array}{c}\hfill \begin{array}{cc}& \xi (us+m(1-u\left)w\right)\le h\left(u\right)\xi \left(s\right)+m(1-h\left(u\right))\xi \left(w\right)-m{l}_{1}u(1-{\varphi }_1){(s-w)}^2.\hfill \end{array}\end{array}$$

Thus, $\xi$ is a SM $(h,m)$-CF. □

The existence of Theorem 1 is monitored by the following example:

Example 3. 

Assuming, $\xi \left(u\right)=u^2,$ ${\varphi }_1=1,$ ${\varphi }_2=2,$ ${\varphi }_3=3,$ $u={\displaystyle \frac{1}{2}}$, $m={\displaystyle \frac{1}{2}}$, $l_1=1,$ and $h\left(u\right)=u^2$ in inequality (5), we obtain

$$\begin{array}{c}\hfill \xi \left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{3}{4}}\right)h\left(2\right)\le h\left(1\right)\xi \left(1\right)+{\displaystyle \frac{1}{2}}(h\left(2\right)-h\left(1\right))\xi \left(3\right)-{\displaystyle \frac{h\left(2\right)}{2}}.\end{array}$$

Hence,

$$6.25\le 12.5.$$

5. Hermite–Hadamard Inequalities

In this section, the H-H inequalities are proved for the proposed class of convexity.

Theorem 2. 

Let $\xi :[0,b]\to \mathbb{R}$ with $b>0$ be a SM $(h,m)$-CF such that for ${\varphi }_1,{\varphi }_2\in [0,b]$ we have ${\varphi }_1<{\varphi }_2$, then

$$\begin{array}{cc}\hfill \xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)+{\displaystyle \frac{m{l}_1}{4}}{F}_u& \le h(1/2)\underset{{\varphi }_1}{\overset{m{\varphi }_2}{\int }}{\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-{\varphi }_1}}ds+m^2(1-h(1/2))\underset{{\displaystyle \frac{{\varphi }_1}{m}}}{\overset{{\varphi }_2}{\int }}{\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-{\varphi }_1}}ds\hfill \\ \hfill & \le m(1-h(1/2)\underset{0}{\overset{1}{\int }}\left(m(1-h\left(u\right))\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)+h\left(u\right)\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\displaystyle \frac{{\varphi }_1}{m^2}}-{\varphi }_2)}^2\right)du\hfill \\ \hfill & +h(1/2)\underset{0}{\overset{1}{\int }}\left(h\left(u\right)\xi \left({\varphi }_1\right)+m(1-h\left(u\right))\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2\right)du,\hfill \end{array}$$

(8)

where

$$F_u=\underset{0}{\overset{1}{\int }}{\left(\left(u({\varphi }_1-{\varphi }_2)\right)+\left((1-u)(m{\varphi }_2-{\displaystyle \frac{{\varphi }_1}{m}})\right)\right)}^{2}du.$$

Proof. 

For $s,w\in [0,b]$, with $b>0$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \xi (us+m(1-u\left)w\right)\le h\left(u\right)\xi \left(s\right)+m(1-h\left(u\right))\xi \left(w\right)-m{l}_{1}u(1-u){(s-w)}^2.\end{array}\end{array}$$

(9)

Put $u=1/2$ in (9) to obtain

$$\begin{array}{c}\hfill \xi \left({\displaystyle \frac{s+mw}{2}}\right)\le h(1/2)\xi \left(s\right)+m(1-h(1/2))\xi \left(w\right)-{\displaystyle \frac{m{l}_1}{4}}{(s-w)}^2.\end{array}$$

(10)

Put $s=u{\varphi }_1+m(1-u){\varphi }_2$ and ${\displaystyle w=(1-u)\frac{{\varphi }_1}{m}+u{\varphi }_2}$ in (10) to obtain

$$\begin{array}{cc}\hfill \xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)& \le h(1/2)\xi (u{\varphi }_1+m(1-u){\varphi }_2)+m(1-h(1/2))\xi ((1-u){\displaystyle \frac{{\varphi }_1}{m}}+u{\varphi }_2)\hfill \\ \hfill & -{\displaystyle \frac{m{l}_1}{4}}{\left(\left(u({\varphi }_1-{\varphi }_2)\right)+\left((1-u)(m{\varphi }_2-{\displaystyle \frac{{\varphi }_1}{m}})\right)\right)}^2.\hfill \end{array}$$

(11)

By integrating (11) with respect to u from 0 to 1, one obtains

$$\begin{array}{cc}\hfill \xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)\underset{0}{\overset{1}{\int }}1du& \le h(1/2)\underset{0}{\overset{1}{\int }}\xi (u{\varphi }_1+m(1-u){\varphi }_2)du+m(1-h(1/2))\underset{0}{\overset{1}{\int }}\xi ((1-u){\displaystyle \frac{{\varphi }_1}{m}}+u{\varphi }_2)du\hfill \\ \hfill & -{\displaystyle \frac{m{l}_1}{4}}\underset{0}{\overset{1}{\int }}{\left(\left(u({\varphi }_1-{\varphi }_2)\right)+\left((1-u)(m{\varphi }_2-{\displaystyle \frac{{\varphi }_1}{m}})\right)\right)}^{2}du.\hfill \end{array}$$

$$\begin{array}{cc}\hfill \xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)& \le h(1/2)\underset{0}{\overset{1}{\int }}\xi (u{\varphi }_1+m(1-u){\varphi }_2)du\hfill \\ \hfill & +m(1-h(1/2))\underset{0}{\overset{1}{\int }}\xi ((1-u){\displaystyle \frac{{\varphi }_1}{m}}+u{\varphi }_2)du-{\displaystyle \frac{m{l}_1}{4}}{F}_u.\hfill \end{array}$$

$$\begin{array}{cc}\hfill \xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)+{\displaystyle \frac{m{l}_1}{4}}{F}_u& \le h(1/2)\underset{0}{\overset{1}{\int }}\xi (u{\varphi }_1+m(1-u){\varphi }_2)du\hfill \\ \hfill & +m(1-h(1/2))\underset{0}{\overset{1}{\int }}\xi ((1-u){\displaystyle \frac{{\varphi }_1}{m}}+u{\varphi }_2)du.\hfill \end{array}$$

(12)

Put $s=u{\varphi }_1+m(1-u){\varphi }_2$ in the first integral of (12), and ${\displaystyle s=(1-u)\frac{{\varphi }_1}{m}+u{\varphi }_2}$ in the second integral of (12), to obtain

$$\begin{array}{c}\hfill ⟹\xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)+{\displaystyle \frac{m{l}_1}{4}}{F}_u\le h(1/2)\underset{{\varphi }_1}{\overset{m{\varphi }_2}{\int }}{\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-{\varphi }_1}}ds+m^2(1-h(1/2))\underset{{\displaystyle \frac{{\varphi }_1}{m}}}{\overset{{\varphi }_2}{\int }}{\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-{\varphi }_1}}ds.\end{array}$$

(13)

By comparing the right-hand sides of (12) and (13), one obtains

$$\begin{array}{ccc}& & h(1/2)\underset{{\varphi }_1}{\overset{m{\varphi }_2}{\int }}{\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-{\varphi }_1}}ds+m^2(1-h(1/2))\underset{{\displaystyle \frac{{\varphi }_1}{m}}}{\overset{{\varphi }_2}{\int }}{\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-{\varphi }_1}}ds\hfill \\ & & =h(1/2)\underset{0}{\overset{1}{\int }}\xi (u{\varphi }_1+m(1-u){\varphi }_2)du+m(1-h(1/2))\underset{0}{\overset{1}{\int }}\xi ((1-u){\displaystyle \frac{{\varphi }_1}{m}}+u{\varphi }_2)du.\hfill \end{array}$$

Since $\xi$ is a SM $(h,m)$-CF, therefore

$$\begin{array}{cc}\hfill & h(1/2)\underset{{\varphi }_1}{\overset{m{\varphi }_2}{\int }}{\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-{\varphi }_1}}ds+m^2(1-h(1/2))\underset{{\displaystyle \frac{{\varphi }_1}{m}}}{\overset{{\varphi }_2}{\int }}{\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-{\varphi }_1}}ds\hfill \\ \hfill & \le h(1/2)\underset{0}{\overset{1}{\int }}\left(h\left(u\right)\xi \left({\varphi }_1\right)+m(1-h\left(u\right))\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2\right)du\hfill \\ \hfill & +m(1-h(1/2)\underset{0}{\overset{1}{\int }}\left(m(1-h\left(u\right))\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)+h\left(u\right)\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\displaystyle \frac{{\varphi }_1}{m^2}}-{\varphi }_2)}^2\right).\hfill \end{array}$$

(14)

From (13) and (14), we obtain

$$\begin{array}{cc}\hfill \xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)+{\displaystyle \frac{m{l}_1}{4}}{F}_u& \le h(1/2)\underset{{\varphi }_1}{\overset{m{\varphi }_2}{\int }}{\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-{\varphi }_1}}ds+m^2(1-h(1/2))\underset{{\displaystyle \frac{{\varphi }_1}{m}}}{\overset{{\varphi }_2}{\int }}{\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-{\varphi }_1}}ds\hfill \\ \hfill & \le h(1/2)\underset{0}{\overset{1}{\int }}\left(h\left(u\right)\xi \left({\varphi }_1\right)+m(1-h\left(u\right))\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2\right)du\hfill \\ \hfill & +m(1-h(1/2)\underset{0}{\overset{1}{\int }}\left(m(1-h\left(u\right))\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)+h\left(u\right)\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\displaystyle \frac{{\varphi }_1}{m^2}}-{\varphi }_2)}^2\right).\hfill \end{array}$$

 □

The next remark shows that Theorem 2 is the generalization of already existing results in the literature.

Remark 2. 

(a) 

By assuming $h\left(u\right)=u$ and $m=1$ in (8), one obtains Theorem 6 of [25];

(b) 

By assuming $l_1=0$ and $m=1$ in (8), one obtains Theorem 3 of [24];

(c) 

By assuming $l_1=0$, $m=1$ and $h\left(u\right)=u$ in (8), we obtain H-H inequalities for CFs [10].

Example 4. 

The existence of H-H inequalities established in Theorem 2 is presented in Figure 3.

In Theorem 3, H-H inequalities are established involving R-L integrals for the SM $(h,m)$-CF.

Theorem 3. 

Assume $\xi :[0,b]\to \mathbb{R}$ is a SM $(h,m)$-CF such that for ${\varphi }_1,{\varphi }_2\in [0,b]$ we have ${\varphi }_1<{\varphi }_2$. Then

$$\begin{array}{cc}\hfill \xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)+{\displaystyle \frac{m{l}_1\beta }{4}}{F}_{\beta }& \le {\displaystyle \frac{\Gamma (\beta +1)}{2{(m{\varphi }_2-{\varphi }_1)}^{\beta }}}\left(M_{{\varphi }_1^{+}}^{\beta }\xi \left(m{\varphi }_2\right)+m^{\beta +1}{M}_{{\varphi }_2^{-}}^{\beta }\xi \left({\displaystyle \frac{{\varphi }_1}{m}}\right)\right)\hfill \\ \hfill & \le {\displaystyle \frac{\beta }{2}}\left(\xi \left({\varphi }_1\right)-m^2\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\right){\int }_0^{1}h\left(u\right)r^{\beta -1}du+{\displaystyle \frac{m}{2}}\left(\xi \left({\varphi }_2\right)+m\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\right)\hfill \\ \hfill & -{\displaystyle \frac{m{l}_1\beta \left({({\varphi }_1-{\varphi }_2)}^2+m{\left(\frac{{\varphi }_1}{m^2}-{\varphi }_2\right)}^2\right)}{2(\beta +1)(\beta +2)}}.\hfill \end{array}$$

(15)

where

$$F_{\beta }=\underset{0}{\overset{1}{\int }}\left({\left(u\right)}^{\beta -1}\times {\left(u({\varphi }_1-{\varphi }_2)+(1-u)(m{\varphi }_2-{\displaystyle \frac{{\varphi }_1}{m}})\right)}^2\right)du.$$

Proof. 

Since $\xi$ is a SM $(h,m)$-CF, therefore

$$\xi ((1-u)s+muw)\le (1-h\left(u\right))\xi \left(s\right)+mh\left(u\right)\xi \left(w\right)-m{l}_{1}u(1-u){(s-w)}^2.$$

(16)

By choosing $u=1/2$ in (16), we obtain

$$\xi \left({\displaystyle \frac{s+mw}{2}}\right)\le (1-h(1/2))\xi \left(s\right)+mh(1/2)\xi \left(w\right)-{\displaystyle \frac{m{l}_1}{4}}{(s-w)}^2.$$

(17)

Assume $s=(u{\varphi }_1+m(1-u){\varphi }_2)$ and ${\displaystyle l=((1-u)\frac{{\varphi }_1}{m}+u{\varphi }_2)}$ in (17) to obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)\le (1-h(1/2))\xi (u{\varphi }_1+m(1-u){\varphi }_2)\hfill \\ & +mh(1/2)\xi ((1-u){\displaystyle \frac{{\varphi }_1}{m}}+u{\varphi }_2)-{\displaystyle \frac{m{l}_1}{4}}{\left(u({\varphi }_1-{\varphi }_2)+(1-u)(m{\varphi }_2-{\displaystyle \frac{{\varphi }_1}{m}})\right)}^2.\hfill \end{array}\end{array}$$

(18)

Multiplying (18) with $u^{\beta -1}$, and then integrating it from 0 to 1 with respect to u, to obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \underset{0}{\overset{1}{\int }}\left({\left(u\right)}^{\beta -1}\times \xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)\right)du\le (1-h(1/2))\underset{0}{\overset{1}{\int }}({\left(u\right)}^{\beta -1}\times \xi (u{\varphi }_1+m(1-u){\varphi }_2))du\hfill \\ & +mh(1/2)\underset{0}{\overset{1}{\int }}({\left(u\right)}^{\beta -1}\times \xi ((1-u){\displaystyle \frac{{\varphi }_1}{m}}+u{\varphi }_2))du-{\displaystyle \frac{m{l}_1}{4}}\underset{0}{\overset{1}{\int }}{(\left(u\right)}^{\beta -1}\times {\left(u({\varphi }_1-{\varphi }_2)+(1-u)(m{\varphi }_2-{\displaystyle \frac{{\varphi }_1}{m}})\right)}^{2}du\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}& ⟹{\displaystyle \frac{1}{\beta }}\left(\xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)\right)\le (1-h(1/2))\underset{0}{\overset{1}{\int }}({\left(u\right)}^{\beta -1}\times \xi (u{\varphi }_1+m(1-u){\varphi }_2))du\hfill \\ & +mh(1/2)\underset{0}{\overset{1}{\int }}({\left(u\right)}^{\beta -1}\times \xi ((1-u){\displaystyle \frac{{\varphi }_1}{m}}+u{\varphi }_2))du-{\displaystyle \frac{m{l}_1}{4}}{F}_{\beta }\hfill \end{array}\end{array}$$

$$\begin{array}{ccc}\hfill & & ⟹{\displaystyle \frac{1}{\beta }}\left(\xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)\right)+{\displaystyle \frac{m{l}_1}{4}}{F}_{\beta }\le (1-h(1/2))\underset{0}{\overset{1}{\int }}({\left(u\right)}^{\beta -1}\times \xi (u{\varphi }_1+m(1-u){\varphi }_2))du\hfill \\ & & +mh(1/2)\underset{0}{\overset{1}{\int }}({\left(u\right)}^{\beta -1}\times \xi ((1-u){\displaystyle \frac{{\varphi }_1}{m}}+u{\varphi }_2))du.\hfill \end{array}$$

(19)

Use $s=(u{\varphi }_1+m(1-u){\varphi }_2),$ in the first integral of (19), and ${\displaystyle s=((1-u)\frac{{\varphi }_1}{m}+u{\varphi }_2),}$ in the second integral of (19), to obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\beta }}\left(\xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)\right)+{\displaystyle \frac{m{l}_1}{4}}{F}_{\beta }& \le (1-h(1/2))\underset{{\varphi }_1}{\overset{m{\varphi }_2}{\int }}\left({\left({\displaystyle \frac{m{\varphi }_2-s}{m{\varphi }_2-{\varphi }_1}}\right)}^{\beta -1}\times {\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-m_1}}\right)ds\hfill \\ \hfill & +m^{2}h(1/2)\underset{{\displaystyle \frac{{\varphi }_1}{m}}}{\overset{{\varphi }_2}{\int }}\left({\left({\displaystyle \frac{ms-{\varphi }_1}{m{\varphi }_2-{\varphi }_1}}\right)}^{\beta -1}\times {\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-{\varphi }_1}}\right)ds.\hfill \end{array}$$

(20)

Since,

$$\begin{array}{cc}\hfill & M_{{\varphi }_1^{+}}^{\beta }\xi \left({\varphi }_2\right)={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}\underset{{\varphi }_1}{\overset{{\varphi }_2}{\int }}{({\varphi }_2-s)}^{\beta -1}\xi \left(s\right)ds,\hfill \\ \hfill & M_{{\varphi }_2^{-}}^{\beta }\xi \left({\varphi }_1\right)={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}\underset{{\varphi }_1}{\overset{{\varphi }_2}{\int }}{(s-{\varphi }_1)}^{\beta -1}\xi \left(s\right)ds.\hfill \end{array}$$

Therefore, (20) becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\beta }}\left(\xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)\right)+{\displaystyle \frac{m{l}_1}{4}}{F}_{\beta }\le {\displaystyle \frac{\Gamma \left(\beta \right)}{{(m{\varphi }_2-{\varphi }_1)}^{\beta }}}\left((1-h(1/2))M_{{\varphi }_1^{+}}^{\beta }\xi \left(m{\varphi }_2\right)+m^{\beta +1}h(1/2)M_{{\varphi }_2^{-}}^{\beta }\xi \left({\displaystyle \frac{{\varphi }_1}{m}}\right)\right).\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)+{\displaystyle \frac{m{l}_1\beta }{4}}{F}_{\beta }\le {\displaystyle \frac{\Gamma (\beta +1)}{2{(m{\varphi }_2-{\varphi }_1)}^{\beta }}}\left(M_{{\varphi }_1^{+}}^{\beta }\xi \left(m{\varphi }_2\right)+m^{\beta +1}{M}_{{\varphi }_2^{-}}^{\beta }\xi \left({\displaystyle \frac{{\varphi }_1}{m}}\right)\right).\end{array}\end{array}$$

(21)

Also, $\xi$ is a SM $(h,m)$-CF, therefore

$$\xi (u{\varphi }_1+m(1-u){\varphi }_2)\le h\left(u\right)\xi \left({\varphi }_1\right)+m(1-h\left(u\right))\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\varphi }_1-{\varphi }_2)}^2,$$

(22)

and

$$\xi \left((1-u){\displaystyle \frac{{\varphi }_1}{m}}+u{\varphi }_2\right)\le m(1-h\left(u\right))\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)+h\left(u\right)\xi \left({\varphi }_2\right)-m{l}_{1}u(1-u){({\displaystyle \frac{{\varphi }_1}{m^2}}-{\varphi }_2)}^2.$$

(23)

Adding (22) and (23) to obtain

$$\begin{array}{ccc}\hfill & & \xi (u{\varphi }_1+m(1-u){\varphi }_2)+m\xi \left((1-u){\displaystyle \frac{{\varphi }_1}{m}}+u{\varphi }_2\right)\hfill \\ & & \le h\left(u\right)\xi \left({\varphi }_1\right)+m(1-h\left(u\right))\xi \left({\varphi }_2\right)+m^2(1-h\left(u\right))\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\hfill \\ & & +mh\left(u\right)\xi \left({\varphi }_2\right)-m{l}_{1}r(1-u){({\varphi }_1-{\varphi }_2)}^2-m^2{l}_{1}r(1-u){({\displaystyle \frac{{\varphi }_1}{m^2}}-{\varphi }_2)}^2\hfill \\ & & =h\left(u\right)\left(\xi \left({\varphi }_1\right)-m^2\xi ({\displaystyle \frac{{\varphi }_1}{m^2}})\right)+m\left(\xi \left({\varphi }_2\right)+m\xi ({\displaystyle \frac{{\varphi }_1}{m^2}})\right)\hfill \\ & & -m{l}_{1}r(1-u){({\varphi }_1-{\varphi }_2)}^2-m^2{l}_{1}r(1-u){({\displaystyle \frac{{\varphi }_1}{m^2}}-{\varphi }_2)}^2.\hfill \end{array}$$

(24)

Multiply (24) with $u^{\beta -1}$ and then integrate from 0 to 1 with respect to u to obtain

$$\begin{array}{ccc}\hfill & & \underset{0}{\overset{1}{\int }}(u^{\beta -1}\times \xi (u{\varphi }_1+m(1-u){\varphi }_2)du+\underset{0}{\overset{1}{\int }}(u^{\beta -1}\times m\xi \left((1-u){\displaystyle \frac{{\varphi }_1}{m}}+r{m}_2\right)du\hfill \\ & & \le \left(\xi \left({\varphi }_1\right)-m^2\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\right){\int }_0^{1}h\left(u\right)u^{\beta -1}du+{\displaystyle \frac{m}{\beta }}\left(\xi \left({\varphi }_2\right)+m\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\right)\hfill \\ & & -m{l}_1{({\varphi }_1-{\varphi }_2)}^2{\int }_0^1{u}^{\beta }(1-u)du-m^2{l}_1{({\displaystyle \frac{{\varphi }_1}{m^2}}-{\varphi }_2)}^2{\int }_0^1{u}^{\beta }(1-u)du.\hfill \end{array}$$

(25)

Use $s=(u{\varphi }_1+m(1-u){\varphi }_2)$ in the first integral of (25) and ${\displaystyle s=((1-u)\frac{{\varphi }_1}{m}+u{\varphi }_2)}$ in the second integral of (25) to obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& \underset{{\varphi }_1}{\overset{m{\varphi }_2}{\int }}\left({\left({\displaystyle \frac{m{\varphi }_2-s}{m{\varphi }_2-{\varphi }_1}}\right)}^{\beta -1}\times {\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-m_1}}\right)ds+m^2\underset{{\displaystyle \frac{{\varphi }_1}{m}}}{\overset{{\varphi }_2}{\int }}\left({\left({\displaystyle \frac{ms-{\varphi }_1}{m{\varphi }_2-{\varphi }_1}}\right)}^{\beta -1}\times {\displaystyle \frac{\xi \left(s\right)}{m{\varphi }_2-{\varphi }_1}}\right)ds\hfill \\ & \le \left(\xi \left({\varphi }_1\right)-m^2\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\right){\int }_0^{1}h\left(u\right)r^{\beta -1}du+{\displaystyle \frac{m}{\beta }}\left(\xi \left({\varphi }_2\right)+m\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\right)\hfill \\ & -{\displaystyle \frac{m{l}_1\left({({\varphi }_1-{\varphi }_2)}^2+m{\left(\frac{{\varphi }_1}{m^2}-{\varphi }_2\right)}^2\right)}{(\beta +1)(\beta +2)}}.\hfill \end{array}\end{array}$$

$$\begin{array}{ccc}& & {\displaystyle \frac{\Gamma \left(\beta \right)}{{(m{\varphi }_2-{\varphi }_1)}^{\beta }}}\left(M_{{\varphi }_1^{+}}^{\beta }\xi \left(m{\varphi }_2\right)+m^{\beta +1}{M}_{{\varphi }_2^{-}}^{\beta }\xi \left({\displaystyle \frac{{\varphi }_1}{m}}\right)\right)\hfill \\ & & \le \left(\xi \left({\varphi }_1\right)-m^2\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\right){\int }_0^{1}h\left(u\right)r^{\beta -1}du+{\displaystyle \frac{m}{\beta }}\left(\xi \left({\varphi }_2\right)+m\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\right)\hfill \\ & & -{\displaystyle \frac{m{l}_1\left({({\varphi }_1-{\varphi }_2)}^2+m{\left(\frac{{\varphi }_1}{m^2}-{\varphi }_2\right)}^2\right)}{(\beta +1)(\beta +2)}}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{\Gamma (\beta +1)}{2{(m{\varphi }_2-{\varphi }_1)}^{\beta }}}\left(M_{{\varphi }_1^{+}}^{\beta }\xi \left(m{\varphi }_2\right)+m^{\beta +1}{M}_{{\varphi }_2^{-}}^{\beta }\xi \left({\displaystyle \frac{{\varphi }_1}{m}}\right)\right)\hfill \\ & & \le {\displaystyle \frac{\beta }{2}}\left(\xi \left({\varphi }_1\right)-m^2\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\right){\int }_0^{1}h\left(u\right)r^{\beta -1}du+{\displaystyle \frac{m}{2}}\left(\xi \left({\varphi }_2\right)+m\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\right)\hfill \\ & & -{\displaystyle \frac{m{l}_1\beta \left({({\varphi }_1-{\varphi }_2)}^2+m{\left(\frac{{\varphi }_1}{m^2}-{\varphi }_2\right)}^2\right)}{2(\beta +1)(\beta +2)}}.\hfill \end{array}$$

(26)

From (21) and (26), one obtains

$$\begin{array}{cc}\hfill \xi \left({\displaystyle \frac{{\varphi }_1+m{\varphi }_2}{2}}\right)+{\displaystyle \frac{m{l}_1\beta }{4}}{F}_{\beta }& \le {\displaystyle \frac{\Gamma (\beta +1)}{2{(m{\varphi }_2-{\varphi }_1)}^{\beta }}}\left(M_{{\varphi }_1^{+}}^{\beta }\xi \left(m{\varphi }_2\right)+m^{\beta +1}{M}_{{\varphi }_2^{-}}^{\beta }\xi \left({\displaystyle \frac{{\varphi }_1}{m}}\right)\right)\hfill \\ \hfill & \le {\displaystyle \frac{\beta }{2}}\left(\xi \left({\varphi }_1\right)-m^2\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\right){\int }_0^{1}h\left(u\right)r^{\beta -1}du+{\displaystyle \frac{m}{2}}\left(\xi \left({\varphi }_2\right)+m\xi \left({\displaystyle \frac{{\varphi }_1}{m^2}}\right)\right)\hfill \\ \hfill & -{\displaystyle \frac{m{l}_1\beta \left({({\varphi }_1-{\varphi }_2)}^2+m{\left(\frac{{\varphi }_1}{m^2}-{\varphi }_2\right)}^2\right)}{2(\beta +1)(\beta +2)}}.\hfill \end{array}$$

 □

The next remark reflects that Theorem 3 is a generalization of already existing results in the literature.

Remark 3. 

(a) 

By assuming $h\left(u\right)=u$, and $l_1=0$ in Theorem 3, one obtains Theorem 2.1 of [31];

(b) 

By putting $h\left(u\right)=u$ in Theorem 3, one obtains Theorem 2.1 of [32];

(c) 

By choosing $h\left(u\right)=u$, $m=1$ and $l_1=0$ in Theorem 3, one obtains Theorem 1.2 of [33];

(d) 

By assuming $m=1$ and $h\left(u\right)=u$ in Theorem 3, one obtains the conclusion for the strongly CFs [30].

Example 5. 

The validity of Theorem 3 is shown in Figure 4.

The next example shows that the established results are more efficient than the existing results in the literature.

Example 6. 

By assuming $u=3/4$, $m=1/2$, $l_1=1$, $\beta =1$, $h\left(u\right)=u^2$ and $\xi \left(u\right)=u^2$, in Theorem 3, we obtain

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{2{\varphi }_1+{\varphi }_2}{4}}\right)}^2+{\displaystyle \frac{1}{8}}{\left({\displaystyle \frac{3({\varphi }_1-{\varphi }_2)}{4}}+{\displaystyle \frac{({\varphi }_2-4{\varphi }_1)}{8}}\right)}^2& \le {\displaystyle \frac{1}{({\varphi }_2-2{\varphi }_1)}}\left({\displaystyle \frac{3{\varphi }_2^3-24{\varphi }_1^3}{24}}\right)\hfill \\ \hfill & \le {\displaystyle \frac{-{\varphi }_1^2}{2}}+{\displaystyle \frac{1}{4}}({\varphi }_2^2+8{\varphi }_1^2)-{\displaystyle \frac{1}{24}}\left({({\varphi }_1-{\varphi }_2)}^2+{\displaystyle \frac{1}{2}}{(4{\varphi }_1-{\varphi }_2)}^2\right).\hfill \end{array}$$

(27)

Choose ${\varphi }_1=2$ in (27) to obtain

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{4+{\varphi }_2}{4}}\right)}^2+{\displaystyle \frac{1}{8}}{\left({\displaystyle \frac{3(2-{\varphi }_2)}{4}}+{\displaystyle \frac{({\varphi }_2-8)}{8}}\right)}^2& \le {\displaystyle \frac{1}{({\varphi }_2-4)}}\left({\displaystyle \frac{3{\varphi }_2^3-192}{24}}\right)\hfill \\ \hfill & \le 2+{\displaystyle \frac{1}{4}}({\varphi }_2^2+32)-{\displaystyle \frac{1}{24}}\left({(2-{\varphi }_2)}^2+{\displaystyle \frac{1}{2}}{(8-{\varphi }_2)}^2\right).\hfill \end{array}$$

(28)

Substitute ${\varphi }_2=16$ in (28) to obtain

$$41.53\le 42\le 60.5.$$

(29)

Similarly, by assuming $r=3/4$, $m=1/2$, $l_1=1$, $\beta =1$, $h\left(u\right)=u^2$ and $\xi \left(u\right)=u^2$, in Theorem 2.1 of [32], we obtain

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{2{\varphi }_1+{\varphi }_2}{4}}\right)}^2+{\displaystyle \frac{1}{8}}\left({\displaystyle \frac{{({\varphi }_1-{\varphi }_2)}^2}{3}}+{\displaystyle \frac{{(\frac{{\varphi }_2}{2}-2{\varphi }_1)}^2}{3}}+{\displaystyle \frac{({\varphi }_1-{\varphi }_2)(\frac{{\varphi }_2}{2}-2{\varphi }_1)}{3}}\right)& \le {\displaystyle \frac{1}{({\varphi }_2-2{\varphi }_1)}}\left({\displaystyle \frac{3{\varphi }_2^3-24{\varphi }_1^3}{24}}\right)\hfill \\ \hfill \le {\displaystyle \frac{-3{\varphi }_1^2}{4}}+{\displaystyle \frac{1}{4}}({\varphi }_2^2+8{\varphi }_1^2)-{\displaystyle \frac{1}{24}}\left({({\varphi }_1-{\varphi }_2)}^2+{\displaystyle \frac{1}{2}}{(4{\varphi }_1-{\varphi }_2)}^2\right).\end{array}$$

(30)

Put ${\varphi }_1=2$ in inequality (30) to obtain

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{4+{\varphi }_2}{4}}\right)}^2+{\displaystyle \frac{1}{8}}\left({\displaystyle \frac{{(2-{\varphi }_2)}^2}{3}}+{\displaystyle \frac{{(\frac{{\varphi }_2}{2}-4)}^2}{3}}+{\displaystyle \frac{(2-{\varphi }_2)(\frac{{\varphi }_2}{2}-4)}{3}}\right)& \le {\displaystyle \frac{1}{({\varphi }_2-4)}}\left({\displaystyle \frac{3{\varphi }_2^3-192}{24}}\right)\hfill \\ \hfill \le -3+{\displaystyle \frac{1}{4}}({\varphi }_2^2+8{\varphi }_1^2)-{\displaystyle \frac{1}{24}}\left({({\varphi }_1-{\varphi }_2)}^2+{\displaystyle \frac{1}{2}}{(4{\varphi }_1-{\varphi }_2)}^2\right).\end{array}$$

(31)

Choose ${\varphi }_2=16$ in inequality (31) to obtain

$$31.5\le 42\le 59.5.$$

(32)

Figure 5 presents the comparison between the inequality (28) and (31).

Remark 4. 

The difference of bounds in Theorem 2 is 18.97, while the difference of bounds in Theorem 2.1 of [32] is 28. This yields that the existing results involving fractional integrals are refined with those having the same fractional integrals but having sharp upper and lower bounds.

6. Conclusions

The paper presents the notion of a SM $(h,m)$-CF which generalizes the theory of a strongly CF [25] and a strongly m-CF [22]. The H-H inequalities demonstrated in [10,24,25,30] can be deduced from the H-H inequalities explored in this paper. Also, the H-H inequalities are incorporated involving R-L integrals, which is a generalization of the results that originated in [30,31,32,33]. Some examples and graphs have been presented to show the existence of this newly defined class of convexity and the established inequalities. Moreover, the comparison of H-H inequalities established by utilizing R-L integrals for the strongly m-CFs and the SM $(h,m)$-CFs is given in Example 6. The numerical and graphical representations of Example 6 show that the present results are more efficient than the ones that already exist in the literature. For our proposed class, we are able to generalize the existing H-H inequalities involving fractional integrals. Other companion inequalities, such as Jensen, Hardy, and Fejér inequalities, can also be studied along with their applications’ counterparts involving fractional integrals for the proposed generalized class. We believe that by accepting these wider viewpoints, more general conclusions can be found, which will benefit fractional calculus and the theory of inequalities.