Fractional Calculus and Confluent Hypergeometric Function Applied in the Study of Subclasses of Analytic Functions

Abstract

The study done for obtaining the original results of this paper involves the fractional integral of the confluent hypergeometric function and presents its new applications for introducing a certain subclass of analytic functions. Conditions for functions to belong to this class are determined and the class is investigated considering aspects regarding coefficient bounds as well as partial sums of these functions. Distortion properties of the functions belonging to the class are proved and radii estimates are established for starlikeness and convexity properties of the class.

1. Introduction

Fractional calculus is used in many investigations lately due to its numerous applications in diverse fields of research. Recently published thorough review articles [1,2] discuss the history of fractional calculus and provide references to its many applications in science and engineering.

This powerful tool applied on special functions provides means for obtaining interesting outcomes. Fractional integral was investigated in its relation to Mittag-Leffler functions by many authors (see, for example, [3,4,5]), connected to Bessel functions and to different operators [6,7].

Since the connection between hypergeometric functions and the theory of univalent functions was established in 1985 with the proof of Bieberbach’s conjecture [8], the confluent hypergeometric function was investigated concerning many aspects such as its univalence [9,10,11], its applications on certain classes of univalent functions [12] and it was used in defining new operators [13].

Fractional integral and confluent hypergeometric functions were previously combined and investigated having in view the theories of differential subordination and superordination. The success of the study has inspired us to use the function which resulted from the combination of the two important tools of research in another direction of study which is introducing and studying a new subclass of analytic functions in the unit disc. This approach has not been previously taken by other researchers; hence, the results presented in this paper should generate interest due to the operator used in defining the new class presented in the next section of the paper. Numerous geometric properties of the class are proven and some of them, such as coefficient estimates, starlikeness and convexity, could inspire other directions of study having this class as main objective.

2. Preliminaries

The main definitions and notations related to univalent functions’ theory are first recalled.

Let $U=\{z\in \mathbb{C}:|z|<1\}$ denote the unit disc of the complex plane and $\mathcal{H}\left(U\right)$ the class of analytic functions in the unit disc.

Let

$${\mathcal{A}}_n=\{f\in \mathcal{H}\left(U\right),f\left(z\right)=z+a_{n+1}{z}^{n+1}+\dots ,z\in U\}$$

with ${\mathcal{A}}_1=\mathcal{A}$.

For n a positive integer and $a\in \mathbb{C},$ the class

$$\mathcal{H}[a,n]=\{f\in \mathcal{H}\left(U\right),f\left(z\right)=a+a_n{z}^n+a_{n+1}{z}^{n+1}+\dots ,z\in U\}$$

is defined, with ${\mathcal{H}}_0=\mathcal{H}\left[0,1\right].$

$${\mathcal{S}}^{*}=\left\{f\in \mathcal{A}:\mathrm{Re}\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}>\alpha ,0<\alpha <1\right\}$$

denotes the class of starlike functions of order $\alpha$ and

$$\mathcal{K}\left(\alpha \right)=\left\{f\in \mathcal{A}:\mathrm{Re}\left(\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}+1\right)>\alpha ,0<\alpha <1\right\}$$

denotes the class of convex functions of order $\alpha$. When $\alpha =0$ the classes of starlike functions and convex functions are obtained, respectively.

The subclass of close-to-convex functions is defined as

$$\mathcal{C}=\left\{f\in \mathcal{H}\left(U\right):\exists \phi \in \mathcal{K},\mathrm{Re}\left(\frac{f^{\prime }\left(z\right)}{{\phi }^{\prime }\left(z\right)}\right)>0,z\in U\right\}.$$

It is also said that function f is close-to-convex with respect to function $\phi$.

The fractional integral of order $\lambda$, known as the Riemann–Liouville fractional integral, used by Owa and Srivastava [14,15] is defined as:

Definition 1.

([14,15]) The fractional integral of order λ ($\lambda >0$) is defined for a function f by

$$D_z^{-\lambda }f\left(z\right)=\frac{1}{\mathsf{\Gamma }\left(\lambda \right)}{\int }_0^z\frac{f\left(t\right)}{{\left(z-t\right)}^{1-\lambda }}dt,$$

where f is an analytic function in a simply-connected region of the z-plane containing the origin, and the multiplicity of ${\left(z-t\right)}^{\lambda -1}$ is removed by requiring $log\left(z-t\right)$ to be real, when $\left(z-t\right)>0.$

The form of the confluent hypergeometric function can be seen, for example, in [9]:

Definition 2.

([9] (p. 5)) Let a and c be complex numbers with $c\ne 0,-1,-2,...$ and consider

$$\varphi \left(a,c;z\right){=}_1{F}_1\left(a,c;z\right)=1+\frac{a}{c}\frac{z}{1!}+\frac{a\left(a+1\right)}{c\left(c+1\right)}\frac{z^2}{2!}+\dots ,z\in U.$$

(1)

This function is called confluent (Kummer) hypergeometric function, is analytic in $\mathbb{C}$ and satisfies Kummer’s differential equation

$$z{w}^{″}\left(z\right)+\left(c-z\right)w^{\prime }\left(z\right)-aw\left(z\right)=0.$$

The confluent (Kummer) hypergeometric function can be written as

$$\varphi \left(a,c;z\right)=\sum _{k=0}^{\infty }\frac{{\left(a\right)}_k}{{\left(c\right)}_k}\frac{z^k}{k!}=\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)}\frac{z^k}{k!},$$

(2)

where

$${\left(d\right)}_k=\frac{\mathsf{\Gamma }\left(d+k\right)}{\mathsf{\Gamma }\left(d\right)}=d\left(d+1\right)\left(d+2\right)\dots \left(d+k-1\right)\mathrm{and}{\left(d\right)}_0=1.$$

In [16], the fractional integral of the confluent hypergeometric function was used for introducing the following operator:

Definition 3.

([6]) Let a and c be complex numbers with $c\ne 0,-1,-2,$ … and $\lambda >0.$ We define the fractional integral of confluent hypergeometric function

$$D_z^{-\lambda }\varphi \left(a,c;z\right)=\frac{1}{\mathsf{\Gamma }\left(\lambda \right)}{\int }_0^z\frac{\varphi \left(a,c;t\right)}{{\left(z-t\right)}^{1-\lambda }}dt=$$

(3)

$$\frac{1}{\mathsf{\Gamma }\left(\lambda \right)}\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(k+1\right)}{\int }_0^z\frac{t^k}{{\left(z-t\right)}^{1-\lambda }}dt.$$

The fractional integral of the confluent hypergeometric function can be written

$$D_z^{-\lambda }\varphi \left(a,c;z\right)=\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\sum _{k=0}^{\infty }\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}{z}^{k+\lambda },$$

(4)

after a simple calculation. Evidently, $D_z^{-\lambda }\varphi \left(a,c;z\right)\in \mathcal{H}\left[0,\lambda \right].$

When the convolution product of the fractional integral of a confluent hypergeometric function with an analytic function resulted in an intriguing operator in article [17], a novel technique was employed.

Definition 4.

([17]) Denote by $D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)$ the operator given by the Hadamard product (the convolution product) of the fractional integral of the confluent hypergeometric function and the analytic function $f\in \mathcal{A}$, $D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right):\mathcal{A}\to \mathcal{A}$,

$$D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)=D_z^{-\lambda }\varphi \left(a,c;z\right)*f\left(z\right).$$

Remark 1.

([17]) If $f\in \mathcal{A}$, $f\left(z\right)=z+{\sum }_{k=2}^{\infty }{a}_k{z}^k,$ then$D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)=z+\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\sum _{k=2}^{\infty }\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}{a}_{k+\lambda }{z}^{k+\lambda }.$

Using the operator given in Definition 4 and inspired by the results published in [18], a new subclass of functions is introduced as follows:

Definition 5.

For $\mu \ge 0,$$\lambda \in \mathbb{N}$, $\gamma \in \mathbb{C}-\left\{0\right\},$$0<\beta \le 1,$$a,c\in \mathbb{C},$$c\ne 0,-1,-2,\dots$, let $\mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$ be the subclass of $\mathcal{A}$ consisting of functions that satisfies the inequality

$$\left|\frac{\lambda \left(1-\mu \right)\frac{D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)}{z}+\mu {\left(D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)\right)}^{\prime }}{\lambda \left(1-\mu \right)\frac{D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)}{z}+\mu {\left(D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)\right)}^{\prime }-\gamma }\right|<\beta .$$

(5)

This class is investigated in the next sections of the paper. Section 3 presents original results regarding conditions for functions $f\in \mathcal{A}$ to be in class $\mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$ and inequalities involving coefficients of the functions from this class emerge as corollary. Conditions for partial sums of functions to remain in the class are also given in this section. Distortion bounds for the functions in class $\mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$ are established in Section 4 and outcome of the studies related to starlikeness and convexity of the functions from the class is revealed in Section 5.

3. Coefficient Estimates and Partial Sums

The study on class $\mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$ begins with giving necessary and sufficient conditions for membership of functions $f\in \mathcal{A}$ to the class.

Theorem 1.

Consider a function $f\in \mathcal{A}$. Function f is said to be in class $\mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$ if and only if function f satisfies:

$$\sum _{k=2}^{\infty }\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}\left(\lambda +k\mu \right)a_{k+\lambda }<\frac{\beta \left|\gamma \right|\mathsf{\Gamma }\left(a\right)}{\left(\beta +1\right)\mathsf{\Gamma }\left(c\right)}-\frac{\left|\lambda +\mu -\lambda \mu \right|\mathsf{\Gamma }\left(a\right)}{\mathsf{\Gamma }\left(c\right)}.$$

(6)

The result is sharp for the function F defined by

$$F\left(z\right)=z+\frac{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{z}^{k+\lambda },k\ge 2.$$

(7)

Proof. 

Suppose that function f satisfies (6). Then, for $\left|z\right|<1$, we have

$$\left|\lambda \left(1-\mu \right)\frac{D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)}{z}+\mu {\left(D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)\right)}^{\prime }\right|-$$

$$\beta \left|\lambda \left(1-\mu \right)\frac{D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)}{z}+\mu {\left(D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)\right)}^{\prime }-\gamma \right|=$$

$$\left|\left(\lambda +\mu -\lambda \mu \right)+\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\sum _{k=2}^{\infty }\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}\left(\lambda +k\mu \right)a_{k+\lambda }{z}^{k+\lambda -1}\right|-$$

$$\beta \left|\left(\lambda +\mu -\lambda \mu \right)+\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\sum _{k=2}^{\infty }\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}\left(\lambda +k\mu \right)a_{k+\lambda }{z}^{k+\lambda -1}-\gamma \right|\le$$

$$\left|\lambda +\mu -\lambda \mu \right|+\left|\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\sum _{k=2}^{\infty }\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}\left(\lambda +k\mu \right)a_{k+\lambda }{z}^{k+\lambda -1}\right|-$$

$$\beta \left|\gamma \right|+\beta \left|\lambda +\mu -\lambda \mu \right|+\beta \left|\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\sum _{k=2}^{\infty }\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}\left(\lambda +k\mu \right)a_{k+\lambda }{z}^{k+\lambda -1}\right|<$$

$$\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|-\beta \left|\gamma \right|+\frac{\left(\beta +1\right)\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\sum _{k=2}^{\infty }\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}\left(\lambda +k\mu \right)a_{k+\lambda }{z}^{k+\lambda -1}<0$$

Hence, by using the maximum modulus Theorem and (5), $f\in \mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$. Conversely, assume that

$$\left|\frac{\lambda \left(1-\mu \right)\frac{D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)}{z}+\mu {\left(D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)\right)}^{\prime }}{\lambda \left(1-\mu \right)\frac{D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)}{z}+\mu {\left(D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)\right)}^{\prime }-\gamma }\right|=$$

$$\left|\frac{\left(\lambda +\mu -\lambda \mu \right)+\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\underset{k=2}{\overset{\infty }{\Sigma }}\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}\left(\lambda +k\mu \right)a_{k+\lambda }{z}^{k+\lambda -1}}{\left(\lambda +\mu -\lambda \mu \right)+\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\underset{k=2}{\overset{\infty }{\Sigma }}\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}\left(\lambda +k\mu \right)a_{k+\lambda }{z}^{k+\lambda -1}-\gamma }\right|<\beta ,z\in U.$$

Since $Re\left(z\right)\le \left|z\right|$ for all $z\in U$, we have

$$Re\left\{\frac{\left(\lambda +\mu -\lambda \mu \right)+\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\underset{k=2}{\overset{\infty }{\Sigma }}\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}\left(\lambda +k\mu \right)a_{k+\lambda }{z}^{k+\lambda -1}}{\left(\lambda +\mu -\lambda \mu \right)+\frac{\mathsf{\Gamma }\left(c\right)}{\mathsf{\Gamma }\left(a\right)}\underset{k=2}{\overset{\infty }{\Sigma }}\frac{\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}\left(\lambda +k\mu \right)a_{k+\lambda }{z}^{k+\lambda -1}-\gamma }\right\}<\beta .$$

(8)

By choosing values of z on the real axis so that $\lambda \left(1-\mu \right)\frac{D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)}{z}+\mu {\left(D_z^{-\lambda }\mathsf{\Phi }\left(a,c\right)\left(z\right)\right)}^{\prime }$ is real and letting $z\to 1$ through real values, we obtain the desired inequality (6). □

Corollary 1.

If function f belongs to class $\mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$, then the following inequality holds:

$$a_{k+\lambda }\le \frac{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)},k\ge 2,$$

(9)

with equality only for functions defined by (7).

Theorem 2.

Let $f_1\left(z\right)=z$ and

$$f_k\left(z\right)=z-\frac{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{z}^{k+\lambda },k\ge 2,$$

(10)

for $\mu \ge 0,$$\lambda \in \mathbb{N}$, $\gamma \in \mathbb{C}-\left\{0\right\},\ge$$0<\beta \le 1,\ge$$a,c\in \mathbb{C},\ge$$c\ne 0,-1,-2,\dots$. Then, f is in the class $\mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$ if and only if it can be expressed in the form

$$f\left(z\right)=\sum _{k=1}^{\infty }{\omega }_k{f}_k\left(z\right),$$

(11)

where ${\omega }_k\ge 0$ and ${\sum }_{k=1}^{\infty }{\omega }_k=1.$

Proof. 

Suppose f can be written as in (11). Then,

$$f\left(z\right)=z-\sum _{k=2}^{\infty }{\omega }_k\frac{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{z}^{k+\lambda }.$$

Now,

$$\sum _{k=2}^{\infty }\frac{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}{\omega }_k$$

$$\frac{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}=\sum _{k=2}^{\infty }{\omega }_k=1-{\omega }_1\le 1.$$

Thus,

$$f\in \mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c).$$

Conversely, let $f\in \mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$. Then, by using (9), setting

$${\omega }_k=\frac{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{a}_{k+\lambda },k\ge 2$$

and ${\omega }_1=1-{\sum }_{k=2}^{\infty }{\omega }_k$, we have $f\left(z\right)={\sum }_{k=1}^{\infty }{\omega }_k{f}_k\left(z\right)$. This completes the proof of Theorem 2. □

4. Distortion Bounds

In this section, we obtain distortion bounds for the class $\mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$.

Theorem 3.

If $f\in \mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$, then

$$r-\frac{c\left(c+1\right)\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(\lambda +3\right)}{a\left(a+1\right)\left(\beta +1\right)\left(\lambda +2\mu \right)}{r}^2\le \left|f\left(z\right)\right|$$

(12)

$$\le r+\frac{c\left(c+1\right)\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(\lambda +3\right)}{a\left(a+1\right)\left(\beta +1\right)\left(\lambda +2\mu \right)}{r}^2$$

holds if the sequence ${\left\{{\sigma }_k(\mu ,\lambda ,a,c)\right\}}_{k=2}^{\infty }$ is non-decreasing, and

$$1-2\frac{c\left(c+1\right)\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(\lambda +3\right)}{a\left(a+1\right)\left(\beta +1\right)\left(\lambda +2\mu \right)}{r}^2\le \left|f^{\prime }\left(z\right)\right|$$

(13)

$$\le 1+2\frac{c\left(c+1\right)\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(\lambda +3\right)}{a\left(a+1\right)\left(\beta +1\right)\left(\lambda +2\mu \right)}{r}^2$$

holds if the sequence ${\left\{\frac{{\sigma }_k(\mu ,\lambda ,a,c)}{k}\right\}}_{k=2}^{\infty }$ is non- decreasing, where

$${\sigma }_k(\mu ,\lambda ,a,c)=\frac{\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(a+k\right)}{\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}.$$

The bounds in (12) and (13) are sharp, for f given by

$$f\left(z\right)=z+\frac{c\left(c+1\right)\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(\lambda +3\right)}{a\left(a+1\right)\left(\beta +1\right)\left(\lambda +2\mu \right)}{r}^2,z=\pm r.$$

(14)

Proof. 

In view of Theorem 1, we have

$$\sum _{k=2}^{\infty }{a}_{k+\lambda }\le \frac{c\left(c+1\right)\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(\lambda +3\right)}{a\left(a+1\right)\left(\beta +1\right)\left(\lambda +2\mu \right)}.$$

(15)

We obtain

$$\left|z\right|-{\left|z\right|}^2\sum _{k=2}^{\infty }{a}_{k+\lambda }\le \left|f\left(z\right)\right|\le \left|z\right|+{\left|z\right|}^2\sum _{k=2}^{\infty }{a}_{k+\lambda }.$$

Thus,

$$r-\frac{c\left(c+1\right)\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(\lambda +3\right)}{a\left(a+1\right)\left(\beta +1\right)\left(\lambda +2\mu \right)}{r}^2\le \left|f\left(z\right)\right|$$

(16)

$$\le r+\frac{c\left(c+1\right)\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(\lambda +3\right)}{a\left(a+1\right)\left(\beta +1\right)\left(\lambda +2\mu \right)}{r}^2.$$

Hence, (12) follows from (16). Further,

$$\sum _{k=2}^{\infty }\left(k+\lambda \right)a_{k+\lambda }\le \frac{c\left(c+1\right)\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(\lambda +3\right)}{a\left(a+1\right)\left(\beta +1\right)\left(\lambda +2\mu \right)}.$$

Hence, (13) follows from

$$1-r^2\sum _{k=2}^{\infty }\left(k+\lambda \right)a_{k+\lambda }\le \left|f^{\prime }\left(z\right)\right|\le 1+r^2\sum _{k=2}^{\infty }\left(k+\lambda \right)a_{k+\lambda }.$$

□

5. Radius of Starlikeness and Convexity

The radii of close-to-convexity, starlikeness and convexity for the class $\mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$ are given in this section.

Theorem 4.

Let the function $f\in \mathcal{A}$ belong to the class $\mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$. Then, f is close-to-convex of order $\delta ,$$0\le \delta <1$ in the disc $\left|z\right|<r$, where

$$r:=\underset{k\ge 2}{inf}\sqrt{\frac{\left(1-\delta \right)\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\left(k+\lambda \right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}}.$$

(17)

The result is sharp, with extremal function f given by (7).

Proof. 

For given $f\in \mathcal{A}$, we must show that

$$\left|f^{\prime }\left(z\right)-1\right|<1-\delta .$$

(18)

By a simple calculation, we have

$$\left|f^{\prime }\left(z\right)-1\right|\le \sum _{k=2}^{\infty }\left(k+\lambda \right)a_{k+\lambda }{\left|z\right|}^2.$$

The last expression is less than $1-\delta$ if

$$\sum _{k=2}^{\infty }\frac{\left(k+\lambda \right)}{1-\delta }{a}_{k+\lambda }{\left|z\right|}^2<1.$$

Using the fact that $f\in \mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$ if and only if

$$\sum _{k=2}^{\infty }\frac{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}{a}_{k+\lambda }<1,$$

(18) holds true if

$$\frac{k+\lambda }{1-\delta }{\left|z\right|}^2\le \sum _{k=2}^{\infty }\frac{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}.$$

Or, equivalently,

$${\left|z\right|}^2\le \sum _{k=2}^{\infty }\frac{\left(1-\delta \right)\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\left(k+\lambda \right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)},$$

which completes the proof. □

Theorem 5.

Let $f\in \mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$. Then:

1. f is starlike of order $\delta ,$$0\le \delta <1$, in the disc $\left|z\right|<r_1$ where,

$$r_1=\underset{k\ge 2}{inf}\frac{\left(1-\delta \right)\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\left(k+\delta -2\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)};$$

2. f is convex of order $\delta ,$$0\le \delta <1$, in the disc $\left|z\right|<r_2$ where,

$$r_2=\underset{k\ge 2}{inf}\frac{\left(1-\delta \right)\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{k\left(k-1\right)\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}.$$

Each of these results is sharp for the extremal function f given by (7).

Proof. 

1. For $0\le \delta <1$, we need to show that

$$\left|\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}-1\right|<1-\delta .$$

(19)

We have

$$\left|\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}-1\right|\le \left|\frac{{\sum }_{k=2}^{\infty }(k-1)a_k\left|z\right|}{1+{\sum }_{k=2}^{\infty }{a}_k\left|z\right|}\right|.$$

The last expression is less than $1-\delta$ if

$$\sum _{k=2}^{\infty }\frac{(k+\delta -2)}{1-\delta }{a}_k\left|z\right|<1.$$

Using the fact that $f\in \mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$ if and only if

$$\sum _{k=2}^{\infty }\frac{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}{a}_{k+\lambda }<1.$$

Equation (19) holds true if

$$\frac{k+\delta -2}{1-\delta }\left|z\right|<\frac{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}.$$

Or, equivalently,

$$\left|z\right|<\frac{\left(1-\delta \right)\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\left(k+\delta -2\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)},$$

which yields the starlikeness of the family.

2. Using the fact that f is convex if and only $z{f}^{\prime }$ is starlike, we can prove (2) using a similar way of the proof of (1). The function f is convex if and only if

$$\left|z{f}^{″}\left(z\right)\right|<1-\delta .$$

(20)

We have

$$\left|z{f}^{″}\left(z\right)\right|\le \left|\sum _{k=2}^{\infty }k(k-1)a_k\left|z\right|\right|<1-\delta$$

$$\sum _{k=2}^{\infty }\frac{k(k-1)}{1-\delta }{a}_k\left|z\right|<1.$$

Using the fact that $f\in \mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$ if and only if

$$\sum _{k=2}^{\infty }\frac{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)}{a}_{k+\lambda }<1.$$

Equation (20) holds true if

$$\frac{k\left(k-1\right)}{1-\delta }\left|z\right|<\frac{\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)},$$

or, equivalently,

$$\left|z\right|<\frac{\left(1-\delta \right)\left(\beta +1\right)\left(\lambda +k\mu \right)\mathsf{\Gamma }\left(c\right)\mathsf{\Gamma }\left(a+k\right)}{k\left(k-1\right)\left(\beta \left|\gamma \right|-\left(\beta +1\right)\left|\lambda +\mu -\lambda \mu \right|\right)\mathsf{\Gamma }\left(a\right)\mathsf{\Gamma }\left(c+k\right)\mathsf{\Gamma }\left(\lambda +k+1\right)},$$

which yields the convexity of the family. □

6. Conclusions

The original results presented in this paper refer to the study on the new subclass of analytic functions $\mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$ introduced in Definition 5. First, conditions for certain analytic functions $f\in \mathcal{A}$ to be part of the class are proved and, as corollary, coefficient inequalities are obtained for those functions. Partial sums of functions from class $\mathcal{D}\mathsf{\Phi }(\mu ,\lambda ,\beta ,\gamma ,a,c)$ are investigated and, as a result, conditions for those to remain in the class are obtained. The functions in the class are evaluated regarding distortion properties and also starlikeness and convexity theorems are proved providing radii for close-to-convexity, starlikeness and convexity.

For future investigations, the class could be transformed considering aspects of fuzzy differential subordination and superordination and quantum calculus.