Geometric Properties of Generalized Integral Operators Related to The Miller–Ross Function

Abstract

It is very well-known that the special functions and integral operators play a vital role in the research of applied and mathematical sciences. In this paper, our aim is to present sufficient conditions for the families of integral operators containing the normalized forms of the Miller–Ross functions such that they can be univalent in the open unit disk. Moreover, we find the convexity order of these operators. In proof of results, we use some differential inequalities related with Miller–Ross functions and well-known lemmas. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples.

1. Introduction

Special functions are mathematical functions that lack a precise formal definition, yet they hold significant importance in various fields such as mathematical analysis, physics, functional analysis, and other branches of applied science. Despite their lack of a rigid definition, these functions are widely utilized due to their valuable properties and widespread applicability. Many elementary functions, such as exponential, trigonometric, and hyperbolic functions, are also treated as special functions. The theory of special functions has earned the attention of many researchers throughout the nineteenth century and has been involved in many emerging fields. Indeed, numerous special functions, including the generalized hypergeometric functions, have emerged as a result of solving specific differential equations. These functions have proven to be instrumental in addressing complex mathematical problems, showcasing their remarkable utility in various domains. The geometric properties such as univalence and convexity of special functions and their integral operators are important in complex analysis. Several researchers have dedicated their efforts to investigating integral operators that incorporate special functions such as the Bessel, Lommel, Struve, Wright, and Mittag–Leffler functions. These studies have focused on examining the geometric properties of these operators within various classes of univalent functions. By exploring the interplay between these integral operators and special functions, researchers have deepened our understanding of the behavior and characteristics of univalent functions in different contexts. It is noteworthy that contemporary researchers in the field are actively pursuing the development of novel theoretical methodologies and techniques that combine observational results with various practical applications. Therefore, the primary objective of this paper is to investigate the criteria for univalence and convexity of integral operators that employ Miller–Ross functions.

Let $\mathcal{A}$ denote the class of analytic functions ℏ of the form

$$\hslash \left(\rho \right)=\rho +\sum _{v=2}^{\infty }{a}_v{\rho }^v$$

in the open unit disk $\mathbb{D}=\left\{\rho :\left|\rho \right|<1,\rho \in \mathbb{C}\right\}$ and satisfy the standard normalization condition:

$$\hslash \left(0\right)=0,{\hslash }^{\prime }\left(0\right)=1.$$

We denote by $\mathcal{S}$ the subclass of $\mathcal{A}$ which are also univalent in $\mathbb{D}$. A function $\hslash \in \mathcal{S}$ is convex of order $\delta \left(0\le \delta <1\right)$ if the following condition holds:

$$\Re \left(1+\frac{\rho {\hslash }^{″}\left(\rho \right)}{{\hslash }^{\prime }\left(\rho \right)}\right)>\delta .$$

For $n\in \mathbb{N}: =\left\{1,2,3,\dots \right\}$, define

$${\mathcal{A}}^n: =\left\{({\hslash }_1,{\hslash }_2,\dots ,{\hslash }_n):{\hslash }_v\in \mathcal{A},v=1,2,\dots ,n\right\}.$$

For ${\hslash }_v\in \mathcal{A}$ ($v=1,2,\dots ,n$), the parameters ${\eta }_v,{\zeta }_v\in \mathbb{C}$ ($v=1,2,\dots ,n$) and $\gamma \in \mathbb{C},$ we define the following three integral operators:

$${\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }:{\mathcal{A}}^n⟶\mathcal{A},$$

$${\mathcal{K}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n}:{\mathcal{A}}^n⟶\mathcal{A}$$

and

$${\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }:{\mathcal{A}}^n⟶\mathcal{A}$$

by

$${\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }\left[{\hslash }_1,{\hslash }_2,\dots ,{\hslash }_n\right]\left(\rho \right): ={\left[\gamma {\int }_0^{\rho }{t}^{\gamma -1}\prod _{v=1}^n{\left({\hslash }_v^{\prime }\left(t\right)\right)}^{{\eta }_v}{\left(\frac{{\hslash }_v\left(t\right)}{t}\right)}^{{\zeta }_v}dt\right]}^{1/\gamma },$$

(1)

$${\mathcal{K}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n}\left[{\hslash }_1,{\hslash }_2,\dots ,{\hslash }_n\right]\left(\rho \right): ={\left[\left(1+\sum _{v=1}^n{\eta }_v\right){\int }_0^{\rho }\prod _{v=1}^n{\left({\hslash }_v\left(t\right)\right)}^{{\eta }_{\nu }}{\left(e^{{\hslash }_v\left(t\right)}\right)}^{{\zeta }_v}dt\right]}^{1/\left(1+{\sum }_{v=1}^n{\eta }_v\right)}$$

(2)

and

$${\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }\left[{\hslash }_1,{\hslash }_2,\dots ,{\hslash }_n\right]\left(\rho \right): ={\left[\gamma {\int }_0^{\rho }{t}^{\gamma -1}\prod _{v=1}^n{\left({\hslash }_v^{\prime }\left(t\right)\right)}^{{\eta }_v}{\left(e^{{\hslash }_v\left(t\right)}\right)}^{{\zeta }_v}dt\right]}^{1/\gamma }.$$

(3)

Here, we need to note that, many authors have studied the integral operators (1), (2) and (3) for some specific parameters as follows:

(1)

${\mathcal{J}}_{0,0,\dots ,0;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }\left[{\hslash }_1,{\hslash }_2,\dots ,{\hslash }_n\right]\equiv {\mathcal{F}}_{1/{\zeta }_1,1/{\zeta }_2,\dots ,1/{\zeta }_n;\gamma }$ (Seenivasagan and Breaz [1]; see also [2,3]);

(2)

${\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;0,0,\dots ,0;n;\gamma }\left[{\hslash }_1,{\hslash }_2,\dots ,{\hslash }_n\right]\equiv {\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;0,0,\dots ,0;n;\gamma }\left[{\hslash }_1,{\hslash }_2,\dots ,{\hslash }_n\right]\equiv {\mathcal{H}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;\gamma }$ (Breaz and Breaz [4]);

(3)

${\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;0,0,\dots ,0;n;1}\left[{\hslash }_1,{\hslash }_2,\dots ,{\hslash }_n\right]\equiv {\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;0,0,\dots ,0;n;1}\left[{\hslash }_1,{\hslash }_2,\dots ,{\hslash }_n\right]\equiv {\mathcal{H}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n}$ (Breaz et al. [5]);

(4)

${\mathcal{J}}_{\eta ;0;1;1}\left[\hslash \right]\equiv {\mathcal{H}}_{\eta }$ (Kim and Merkes [6]; see also Pfaltzgraff [7]);

(5)

${\mathcal{L}}_{0;\zeta ;1;\gamma }\left[\hslash \right]\equiv {\mathcal{Q}}_{\zeta }$ (Pescar [8]);

(6)

${\mathcal{K}}_{\eta ,\eta ,\dots ,\eta ;0,0,\dots ,0;n}\left[{\hslash }_1,{\hslash }_2,\dots ,{\hslash }_n\right]\equiv {\mathcal{G}}_{n,\eta }\left[{\hslash }_1,{\hslash }_2,\dots ,{\hslash }_n\right]$ (Breaz and Breaz [9]; see also [10]);

(7)

${\mathcal{K}}_{\eta ;0;1}\left[\hslash \right]\equiv {\mathcal{G}}_{1,\eta }\left[\hslash \right]$ (Moldoveanu and Pascu [11]).

Furthermore, the specific integral operators via an obvious parametric changes of the classical Bessel function $J_{\nu }\left(\rho \right)$ of order $\nu$ and of the first kind by Deniz et al. [12] were introduced and they worked on the univalence condition of the related integral operators. In addition, the starlikeness, convexity and uniform convexity of integral operators containing these equivalent forms of $J_{\nu }\left(\rho \right)$ were discussed by Raza et al. [13] and Deniz [14]. Recently, some sufficient conditions for univalence of various linear fractional derivative operators containing the normalized forms of the similar parametric variation of $J_{\nu }\left(\rho \right)$ were investigated by Al-Khrasani et al. [15]. Moreover, the theory of derivatives and integrals of an arbitrary complex or real order has been utilized not only in complex analysis, but also in the mathematical analysis and modeling of real-world problems in applied sciences (see, for example, [16,17]).

Inspired by the studies mentioned above, in the present paper, we work on some mappings and univalence and convexity conditions for the integral operators given by (1), (2) and (3), related to the following Miller–Ross function $E_{\xi ,\varrho },$ defined by

$$E_{\xi ,\varrho }\left(\rho \right)={\rho }^{\xi }{e}^{\varrho \rho }{\gamma }^{*}\left(\xi ,\varrho \rho \right),$$

where ${\gamma }^{*}$ is the incomplete gamma function (see [18]).

$E_{\xi ,\varrho }\left(\rho \right)$ a solution of the following ordinary differential equation

$$Dy-\varrho y=\frac{{\rho }^{\xi -1}}{\mathsf{\Gamma }\left(\xi \right)},\xi >0.$$

With the help of the gamma function we obtain the following series form of $E_{\xi ,\varrho }\left(\rho \right)$:

$$E_{\xi ,\varrho }\left(\rho \right)={\rho }^{\xi }\sum _{v=0}^{\infty }\frac{{\left(\varrho \rho \right)}^v}{\mathsf{\Gamma }\left(\xi +v+1\right)},$$

where $\varrho ,\rho \in \mathbb{C}.$

The function $E_{\xi ,\varrho }\left(\rho \right)$ does not belong to the class $\mathcal{A}$. The normalization form of the function $E_{\xi ,\varrho }$ is written as

$${\mathbb{E}}_{\xi ,\varrho }\left(\rho \right)=\mathsf{\Gamma }\left(\xi +1\right){\rho }^{1-\xi }{E}_{\xi ,\varrho }\left(\rho \right)=\sum _{v=0}^{\infty }\frac{{\varrho }^v\mathsf{\Gamma }\left(\xi +1\right)}{\mathsf{\Gamma }\left(\xi +v+1\right)}{\rho }^{v+1},$$

(4)

where $\xi >-1$ and $\varrho >0.$

Recently, Eker and Ece [19] and Şeker et al. [20] studied geometric and characteristic properties of this function, respectively. Also, some problems as partial sums, coefficient inequalities, inclusion relations and neighborhoods for Miller-Ross function were studied by Kazımoğlu [21,22].

We note that by choosing particular values for $\xi$ and $\varrho ,$ we obtain the following functions

$${\mathbb{E}}_{1,1/2}\left(\rho \right)=2e^{\rho /2}-2,{\mathbb{E}}_{3,1}\left(\rho \right)=\frac{6e^{\rho }-3{\rho }^2-6\rho -6}{{\rho }^2}$$

and

$${\mathbb{E}}_{\frac{1}{2},1}\left(\rho \right)=\frac{1}{2}{e}^{\rho /5}\sqrt{5\pi }\sqrt{\rho }\mathrm{Erf}\sqrt{\frac{\rho }{5}},{\mathbb{E}}_{2,1/2}\left(\rho \right)=\frac{4\left(2e^{\rho /2}-\rho -2\right)}{\rho },$$

where Erf $\sqrt{\rho }$ is the error function.

Let ${\xi }_v>-1$ for $v=1,2,\dots ,n$ and $\varrho >0.$ Consider the functions ${\mathbb{E}}_{{\xi }_v,\varrho }$ defined by

$${\mathbb{E}}_{{\xi }_v,\varrho }\left(\rho \right)=\mathsf{\Gamma }\left({\xi }_v+1\right){\rho }^{1-{\xi }_v}{E}_{{\xi }_v,\varrho }\left(\rho \right).$$

Using the functions ${\mathbb{E}}_{{\xi }_v,\varrho }$ and the integral operators given by (1), (2) and (3), we define ${\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho },$${\mathcal{K}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n}^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }$ and ${\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }:\mathbb{D}⟶\mathbb{C}$ as follows:

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }\left(\rho \right):=& & {\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }\left[{\mathbb{E}}_{{\xi }_1,\varrho },{\mathbb{E}}_{{\xi }_2,\varrho },\dots ,{\mathbb{E}}_{{\xi }_n,\varrho }\right]\left(\rho \right)\hfill \\ & =& {\left[\gamma {\int }_0^{\rho }{t}^{\gamma -1}\prod _{v=1}^n{\left({\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(t\right)\right)}^{{\eta }_v}{\left(\frac{{\mathbb{E}}_{{\xi }_v,\varrho }\left(t\right)}{t}\right)}^{{\zeta }_v}dt\right]}^{1/\gamma },\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill {\mathcal{K}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n}^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }\left(\rho \right):=& & {\mathcal{K}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n}\left[{\mathbb{E}}_{{\xi }_1,\varrho },{\mathbb{E}}_{{\xi }_2,\varrho },\dots ,{\mathbb{E}}_{{\xi }_n,\varrho }\right]\left(\rho \right)\hfill \\ & =& {\left[\left(1+\sum _{v=1}^n{\eta }_v\right){\int }_0^{\rho }\prod _{v=1}^n{\left({\mathbb{E}}_{{\xi }_v,\varrho }\left(t\right)\right)}^{{\eta }_v}{\left(e^{{\mathbb{E}}_{{\xi }_v,\varrho }\left(t\right)}\right)}^{{\zeta }_v}dt\right]}^{1/\left(1+{\sum }_{v=1}^n{\eta }_v\right)}\hfill \end{array}$$

(6)

and

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }\left(\rho \right):=& & {\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }\left[{\mathbb{E}}_{{\xi }_1,\varrho },{\mathbb{E}}_{{\xi }_2,\varrho },\dots ,{\mathbb{E}}_{{\xi }_n,\varrho }\right]\left(\rho \right)\hfill \\ & =& {\left[\gamma {\int }_0^z{t}^{\gamma -1}\prod _{v=1}^n{\left({\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(t\right)\right)}^{{\eta }_v}{\left(e^{{\mathbb{E}}_{{\xi }_v,\varrho }\left(t\right)}\right)}^{{\zeta }_v}dt\right]}^{1/\gamma }.\hfill \end{array}$$

(7)

An extensive literature in geometric function theory dealing with the geometric properties of the integral operators using different types of special functions can be found. In 2010, some integral operators containing Bessel functions were studied by Baricz and Frasin [2]. They obtained some sufficient conditions for univalence of these operators. The convexity and strongly convexity of the integral operators given in [2] were investigated by Arif and Raza [23] and Frasin [24]. Deniz [14] and Deniz et al. [12] gave convexity and univalence conditions for integral operators involving generalized Bessel Functions, respectively. Between 2018 and 2020, Mahmood et al. [25], Mahmood and his co-authors [26] and Din and Yalçın [27] investigated the certain geometric properties such as univalence, convexity, strongly starlikeness and strongly convexity of integral operators involving Struve functions. Recently, Din and Yalçın [28] obtained some sufficients condidions on starlikeness, convexity and uniformly close-to-convexity of the modified Lommel function. Park et al. [29] investigated univalence and convexity conditions for certain integral operators involving the Lommel function. Srivastava and his co-authors [30] studied sufficient conditions for univalence of certain integral operators involving the normalized Mittag–Leffler functions. Oros [31] studied geometric properties of certain classes of univalent functions using the classical Bernardi and Libera integral operators and the confluent (or Kummer) hypergeometric function. Very recently, Raza et al. [32] obtained the necessary conditions for the univalence of integral operators containing the generalized Bessel function. Studies on this subject are still ongoing.

Motivated by the these works, we obtain some sufficient conditions for the operators (5), (6) and (7), in order to be univalent in $\mathbb{D}$. Moreover, we determine the order of the convexity of these integral operators. By using Mathematica (version 8.0), we give some graphics that support the main results.

2. A Set of Lemmas

The following lemmas will be required in our current research.

Lemma 1 

(see Pescar [33]). Let α and β be complex number such that

$$\Re \left(\alpha \right)>0\mathrm{and}\left|\beta \right|\leqq 1\left(\beta \ne -1\right).$$

If the function $h\in \mathcal{A}$ satisfies the following inequality:

$$\left|\beta {\left|\rho \right|}^{2\alpha }+\left(1-{\left|\rho \right|}^{2\alpha }\right)\frac{\rho h^{″}\left(\rho \right)}{\alpha h^{\prime }\left(\rho \right)}\right|\leqq 1$$

for all $\rho \in \mathbb{D},$ then the function $F_{\alpha }\in \mathcal{A}$ defined by

$$F_{\alpha }\left(\rho \right)=\left(\alpha {\int }_0^{\rho }{t}^{\alpha -1}{h}^{\prime }\left(t\right)dt\right){)}^{1/\alpha }$$

(8)

is in the class $\mathcal{S}.$

Lemma 2 

(see Pascu [34]). Let $\varpi \in \mathbb{C}$ such that $\Re \left(\varpi \right)>0.$ If $h\in \mathcal{A}$ satisfies the following inequality:

$$\left(\frac{1-{\left|\rho \right|}^{2\Re \left(\varpi \right)}}{\Re \left(\varpi \right)}\right)\left|\frac{\rho h^{″}\left(\rho \right)}{h^{\prime }\left(\rho \right)}\right|\leqq 1$$

for all $\rho \in \mathbb{D}.$ Then, for all $\alpha \in \mathbb{C}$ such that

$$\Re \left(\alpha \right)\geqq \Re \left(\varpi \right),$$

the function $F_{\alpha }$ defined by (8) is in the class $\mathcal{S}.$

Lemma 3. 

Let $\xi >-1$ and $\varrho >0.$ Then, for $\forall \rho \in \mathbb{D},$ the function ${\mathbb{E}}_{\xi ,\varrho }$ defined by (4) provides the following inequalities:

$$\left|{\mathbb{E}}_{\xi ,\varrho }^{\prime }\left(\rho \right)-\frac{{\mathbb{E}}_{\xi ,\varrho }\left(\rho \right)}{\rho }\right|\leqq \frac{\varrho \left(\xi +1\right)}{{\left(\xi -\varrho +1\right)}^2}\left(\varrho -1<\xi \right),$$

(9)

$$\left|\frac{\rho {\mathbb{E}}_{\xi ,\varrho }^{\prime }\left(\rho \right)}{{\mathbb{E}}_{\xi ,\varrho }\left(\rho \right)}-1\right|\leqq \frac{\varrho \left(\xi +1\right)}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}\left(2\varrho -1<\xi \right),$$

(10)

$$\left|\rho {\mathbb{E}}_{\xi ,\varrho }^{\prime }\left(\rho \right)\right|\leqq {\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\left(\varrho -1<\xi \right),$$

(11)

$$\left|\frac{\rho {\mathbb{E}}_{\xi ,\varrho }^{\prime }\left(\rho \right)}{{\mathbb{E}}_{\xi ,\varrho }\left(\rho \right)}\right|\leqq \frac{{\left(\xi +1\right)}^2}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}\left(2\varrho -1<\xi \right)$$

(12)

and

$$\left|\frac{\rho {\mathbb{E}}_{\xi ,\varrho }^{″}\left(\rho \right)}{{\mathbb{E}}_{\xi ,\varrho }^{\prime }\left(\rho \right)}\right|\leqq \frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}\left(\left(2+\sqrt{2}\right)\varrho -1<\xi \right).$$

(13)

Proof. 

The inequalities (9) and (10) were proved by Eker and Ece [19]. On the other hand, by using the triangle inequality and the following inequality (see [19])

$$\frac{\mathsf{\Gamma }\left(\xi +1\right)}{\mathsf{\Gamma }\left(\xi +v\right)}\le \frac{1}{{\left(\xi +1\right)}^{v-1}},$$

(14)

we have

$$\left|\frac{{\mathbb{E}}_{\xi ,\varrho }\left(\rho \right)}{\rho }\right|\le \frac{\xi -2\varrho +1}{\xi -\varrho +1}\left(2\varrho -1<\xi \right)$$

(15)

and

$$\begin{array}{ccc}\hfill \left|\rho {\mathbb{E}}_{\xi ,\varrho }^{\prime }\left(\rho \right)\right|& =& \left|\rho +\sum _{v=2}^{\infty }\frac{v\mathsf{\Gamma }\left(\xi +1\right){\varrho }^{v-1}}{\mathsf{\Gamma }\left(\xi +v\right)}{\rho }^v\right|\hfill \\ & \le & 1+\sum _{v=2}^{\infty }\frac{v\mathsf{\Gamma }\left(\xi +1\right){\varrho }^{v-1}}{\mathsf{\Gamma }\left(\xi +v\right)}\hfill \\ & \le & 1+\sum _{v=2}^{\infty }v{\left(\frac{\varrho }{\xi +1}\right)}^{v-1}={\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\left(\varrho -1<\xi \right).\hfill \end{array}$$

(16)

Thus, from (15) and (16), we obtain

$$\left|\frac{\rho {\mathbb{E}}_{\xi ,\varrho }^{\prime }\left(\rho \right)}{{\mathbb{E}}_{\xi ,\varrho }\left(\rho \right)}\right|\leqq \frac{{\left(\xi +1\right)}^2}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}\left(2\varrho -1<\xi \right).$$

Using the inequality (14), it follows that

$$\begin{array}{ccc}\hfill \left|\rho {\mathbb{E}}_{\xi ,\varrho }^{″}\left(\rho \right)\right|& =& \left|\sum _{v=2}^{\infty }\frac{v\left(v-1\right)\mathsf{\Gamma }\left(\xi +1\right){\varrho }^{v-1}}{\mathsf{\Gamma }\left(\xi +v\right)}{\rho }^{v-1}\right|\hfill \\ & \le & \sum _{v=2}^{\infty }\frac{v\left(v-1\right)\mathsf{\Gamma }\left(\xi +1\right){\varrho }^{v-1}}{\mathsf{\Gamma }\left(\xi +v\right)}\hfill \\ & \le & 1+\sum _{v=2}^{\infty }v\left(v-1\right){\left(\frac{\varrho }{\xi +1}\right)}^{v-1}=\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^3}\hfill \end{array}$$

(17)

for $\varrho -1<\xi$. Finally, applying reverse triangle inequality, we conclude that

$$\begin{array}{ccc}\hfill \left|{\mathbb{E}}_{\xi ,\varrho }^{\prime }\left(\rho \right)\right|& =& \left|1+\sum _{v=2}^{\infty }\frac{v\mathsf{\Gamma }\left(\xi +1\right){\varrho }^{v-1}}{\mathsf{\Gamma }\left(\xi +v\right)}{\rho }^{v-1}\right|\hfill \\ & \ge & 1-\sum _{v=2}^{\infty }\frac{v\mathsf{\Gamma }\left(\xi +1\right){\varrho }^{v-1}}{\mathsf{\Gamma }\left(\xi +v\right)}\hfill \\ & \ge & 1-\sum _{v=2}^{\infty }v{\left(\frac{\varrho }{\xi +1}\right)}^{v-1}=\frac{{\left(\xi -\varrho +1\right)}^2-2\xi \varrho -2\varrho +{\varrho }^2}{{\left(\xi -\varrho +1\right)}^2}\hfill \end{array}$$

(18)

for $\left(2+\sqrt{2}\right)\varrho -1<\xi$. Next, by combining the inequalities (17) with (18), we can easily see that

$$\left|\frac{\rho {\mathbb{E}}_{\xi ,\varrho }^{″}\left(\rho \right)}{{\mathbb{E}}_{\xi ,\varrho }^{\prime }\left(\rho \right)}\right|\leqq \frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}\left(\left(2+\sqrt{2}\right)\varrho -1<\xi \right).$$

This completes the proof. □

3. Univalence and Convexity Conditions for the Integral Operator in (5)

Firstly, we take into account the integral operator defined by (5).

Theorem 1. 

Let $v=1,2,\dots ,n,$${\xi }_v>-1,$$\varrho >0$ and $\left(2+\sqrt{2}\right)\varrho -1<{\xi }_v.$ Also, let $\gamma , \beta , {\eta }_v$ and ${\zeta }_v$ be in $\mathbb{C}$ such that

$$\Re \left(\gamma \right)>0,\left|\beta \right|\le 1\left(\beta \ne -1\right).$$

Assume that these numbers satisfy the following inequality:

$$\left|\beta \right|+\frac{1}{\left|\gamma \right|}\left(\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}\sum _{v=1}^n\left|{\eta }_v\right|+\frac{\varrho \left(\xi +1\right)}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}\sum _{v=1}^n\left|{\zeta }_v\right|\right)\leqq 1,$$

where $\xi =min\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_n\right\}.$ Then the function ${\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }$ defined by (5) is in the class $\mathcal{S}.$

Proof. 

Let us define the function $\phi$ as follows:

$$\phi \left(\rho \right)={\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;1}^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }\left(\rho \right)={\int }_0^{\rho }\prod _{v=1}^n{\left({\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(t\right)\right)}^{{\eta }_v}{\left(\frac{{\mathbb{E}}_{{\xi }_v,\varrho }\left(t\right)}{t}\right)}^{{\zeta }_v}dt.$$

(19)

First of all, we observe that $\phi \left(\rho \right)={\phi }^{\prime }\left(\rho \right)-1=0,$ since ${\mathbb{E}}_{{\xi }_v,\varrho }\in \mathcal{A}$ for all $v=1,2,\dots ,n.$ However, we also have

$${\phi }^{\prime }\left(\rho \right)=\prod _{v=1}^n{\left({\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)\right)}^{{\eta }_v}{\left(\frac{{\mathbb{E}}_{{\xi }_v,\varrho }\left(\rho \right)}{\rho }\right)}^{{\zeta }_v}.$$

(20)

Taking the logarithmic derivative of both sides of (20), we get

$$\frac{\rho {\phi }^{″}\left(\rho \right)}{{\phi }^{\prime }\left(\rho \right)}=\sum _{v=1}^n{\eta }_v\frac{\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{\prime \prime }\left(\rho \right)}{{\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)}+\sum _{v=1}^n{\zeta }_v\left(\frac{\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)}{{\mathbb{E}}_{{\xi }_v,\varrho }\left(\rho \right)}-1\right)$$

(21)

and, from (10) and (13), we have

$$\begin{array}{ccc}\hfill \left|\frac{\rho {\phi }^{″}\left(\rho \right)}{{\phi }^{\prime }\left(\rho \right)}\right|& \leqq & \sum _{v=1}^n\left(\left|{\eta }_v\right|\left|\frac{\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{″}\left(\rho \right)}{{\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)}\right|+\left|{\zeta }_v\right|\left|\frac{\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)}{{\mathbb{E}}_{{\xi }_v,\varrho }\left(\rho \right)}-1\right|\right)\hfill \\ & \leqq & \sum _{v=1}^n\left(\left|{\eta }_v\right|\frac{{\left({\xi }_v-\varrho +1\right)}^3+2\varrho {\left({\xi }_v+1\right)}^2}{{\left({\xi }_v-\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2{\xi }_v\varrho +1\right)}+\left|{\zeta }_v\right|\frac{\varrho \left({\xi }_v+1\right)}{\left({\xi }_v-\varrho +1\right)\left({\xi }_v-2\varrho +1\right)}\right)\hfill \\ & \leqq & \sum _{v=1}^n\left(\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\left|{\zeta }_v\right|\frac{\varrho \left(\xi +1\right)}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}\right),\hfill \end{array}$$

(22)

$$\left(\rho \in \mathbb{D};\left(2+\sqrt{2}\right)\varrho -1<\xi ,{\xi }_v<{\varrho }_0,\left(v=1,2,\dots ,n\right)\right)$$

where ${\varrho }_0=\frac{1}{3}\left[-3+\varrho \left(7+\frac{49}{{\left(370-3\sqrt{2139}\right)}^{1/3}}+{\left(370-3\sqrt{2139}\right)}^{1/3}\right)\right].$ Here, we have also used the fact that the functions

$${\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2:\left(\left(2+\sqrt{2}\right)\varrho -1,{\varrho }_0\right)⟶\mathbb{R},$$

defined by

$${\mathsf{\Theta }}_1\left(x\right)=\frac{{\left(x-\varrho +1\right)}^3+2\varrho {\left(x+1\right)}^2}{{\left(x-\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2x\varrho +1\right)}\mathrm{and}{\mathsf{\Theta }}_2\left(x\right)=\frac{\varrho \left(x+1\right)}{\left(x-\varrho +1\right)\left(x-2\varrho +1\right)},$$

are decreasing and, consequently, we have

$$\frac{{\left({\xi }_v-\varrho +1\right)}^3+2\varrho {\left({\xi }_v+1\right)}^2}{{\left({\xi }_v-\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2{\xi }_v\varrho +1\right)}<\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}$$

(23)

and

$$\frac{\varrho \left({\xi }_v+1\right)}{\left({\xi }_v-\varrho +1\right)\left({\xi }_v-2\varrho +1\right)}<\frac{\varrho \left(\xi +1\right)}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}.$$

Therefore, from hypothesis of theorem we obtain

$$\begin{array}{ccc}& & \left|\beta {\left|\rho \right|}^{2\gamma }+\left(1-{\left|\rho \right|}^{2\gamma }\right)\frac{\rho {\phi }^{″}\left(\rho \right)}{\gamma {\phi }^{\prime }\left(\rho \right)}\right|\hfill \\ & & \leqq \left|\beta \right|+\frac{1}{\left|\gamma \right|}\left(\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}\sum _{v=1}^n\left|{\eta }_v\right|+\frac{\varrho \left(\xi +1\right)}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}\sum _{v=1}^n\left|{\zeta }_v\right|\right)\hfill \\ & & \leqq 1,\hfill \end{array}$$

which imply that the function ${\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }\in \mathcal{S}$ by Lemma 1. □

Theorem 2. 

Let the parameters $\varrho ,$$\gamma ,{\eta }_v,$${\xi }_v$ and ${\zeta }_v$$(v=1,2,\dots ,n)$ be as in Theorem 1. Suppose that $\xi =min\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_n\right\}$ and that the following inequality holds true:

$$\Re \left(\gamma \right)\geqq \sum _{v=1}^n\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\sum _{v=1}^n\left|{\zeta }_v\right|\frac{\varrho \left(\xi +1\right)}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}.$$

Then the function ${\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }$ defined by (5) is in the class $\mathcal{S}.$

Proof. 

Let us consider the function $\phi$ as in (19). From (22) and hypothesis of theorem, we get

$$\begin{array}{ccc}& & \frac{1-{\left|\rho \right|}^{2\Re \left(\gamma \right)}}{\Re \left(\gamma \right)}\left|\frac{\rho {\phi }^{″}\left(\rho \right)}{{\phi }^{\prime }\left(\rho \right)}\right|\hfill \\ & & \leqq \frac{1-{\left|\rho \right|}^{2\Re \left(\gamma \right)}}{\Re \left(\gamma \right)}\sum _{v=1}^n\left(\left|{\eta }_v\right|\frac{{\left({\xi }_v-\varrho +1\right)}^3+2\varrho {\left({\xi }_v+1\right)}^2}{{\left({\xi }_v-\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2{\xi }_v\varrho +1\right)}+\left|{\zeta }_v\right|\frac{\varrho \left({\xi }_v+1\right)}{\left({\xi }_v-\varrho +1\right)\left({\xi }_v-2\varrho +1\right)}\right)\hfill \\ & & \leqq \frac{1}{\Re \left(\gamma \right)}\sum _{v=1}^n\left(\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\left|{\zeta }_v\right|\frac{\varrho \left(\xi +1\right)}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}\right)\hfill \\ & & \le 1.\hfill \end{array}$$

(24)

By Lemma 2, the inequality (24) imply that the function ${\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }\in \mathcal{S}.$ □

Theorem 3. 

Let the parameters $\varrho ,$${\eta }_v,$${\xi }_v$ and ${\zeta }_v$$(v=1,2,\dots ,n)$ be as in Theorem 1. Suppose that $\xi =min\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_n\right\}$ and that the following inequality holds true:

$$0<\sum _{v=1}^n\left(\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\left|{\zeta }_v\right|\frac{\varrho \left(\xi +1\right)}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}\right)\leqq 1.$$

Then the function ${\mathcal{J}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;1}^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }$ defined by (5) with $\gamma =1$ is convex of order δ given by

$$\delta =1-\sum _{v=1}^n\left(\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\left|{\zeta }_v\right|\frac{\varrho \left(\xi +1\right)}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}\right).$$

Proof. 

The inequality (22) and hypothesis of theorem show that

$$\begin{array}{ccc}\hfill \left|\frac{\rho {\phi }^{″}\left(\rho \right)}{{\phi }^{\prime }\left(\rho \right)}\right|& \leqq & \sum _{v=1}^n\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\sum _{v=1}^n\left|{\zeta }_v\right|\frac{\varrho \left(\xi +1\right)}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}\hfill \\ & =& 1-\delta .\hfill \end{array}$$

As a result, the function $\phi$ is convex of order

$$\delta =1-\sum _{v=1}^n\left(\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\left|{\zeta }_v\right|\frac{\varrho \left(\xi +1\right)}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}\right).$$

□

In Theorem 1 with $n=1,{\xi }_1=1$ and $\varrho =1/2,$ we can write the following corollary.

Corollary 1. 

Let $\gamma ,\beta ,\eta$ and ζ be in $\mathbb{C}$ such that $\Re \left(\gamma \right)>0$ and $\left|\beta \right|\le 1\left(\beta \ne -1\right).$ If the inequality

$$\left|\beta \right|+\frac{1}{\left|\gamma \right|}\left(\frac{59}{4}\left|\eta \right|+\frac{2}{3}\left|\zeta \right|\right)\leqq 1$$

holds true, then the function

$${\left[\gamma {\int }_0^{\rho }{t}^{-\zeta +\gamma -1}{\left(e^{t/2}\right)}^{\eta }{\left(2e^{t/2}-2\right)}^{\zeta }dt\right]}^{1/\gamma }$$

is in the class $\mathcal{S}.$

Example 1. 

$$f_1\left(\rho \right)={\int }_0^{\rho }{t}^{-1}\left(e^{t/200}\right)\left(2e^{t/2}-2\right)dt\in \mathcal{S}.$$

Normally, it is almost impossible to find the geometric properties (univalent, convex, starlike, etc.) of a complex function and especially integral operators with classical methods. However, from Corollary 1 (also from Figure 1) with $\gamma =\zeta =1$ and $\eta =0.01,$ we can see that the function $f_1$ belongs to the class $\mathcal{S}.$

Figure 1. Image of $\mathbb{D}$ under $f_1$.

Setting $n=1,{\xi }_1=3$ and $\varrho =1$ in the Theorem 1, we can get result below.

Corollary 2. 

Let $\gamma ,\beta ,\eta$ and ζ be in $\mathbb{C}$ such that $\Re \left(\gamma \right)>0$ and $\left|\beta \right|\le 1\left(\beta \ne -1\right).$ If the inequality

$$\left|\beta \right|+\frac{1}{\left|\gamma \right|}\left(\frac{59}{2}\left|\eta \right|+\frac{2}{3}\left|\zeta \right|\right)\leqq 1$$

holds, then the function

$${\left[6^{\eta }{3}^{\zeta }\gamma {\int }_0^{\rho }{t}^{-3\eta -3\zeta +\gamma -1}{\left(t{e}^t-2e^t+t+2\right)}^{\eta }{\left(2e^t-t^2-2t-2\right)}^{\zeta }dt\right]}^{1/\gamma }$$

is in the class $\mathcal{S}.$

From Theorem 3 with $n=1, {\xi }_1=1$ and $\varrho =1/2,$ we can get result below.

Corollary 3. 

Let η and ζ be complex numbers such that

$$\frac{59}{4}\left|\eta \right|+\frac{2}{3}\left|\zeta \right|\leqq 1.$$

Then the function

$${\int }_0^{\rho }{t}^{-\zeta }{\left(e^{t/2}\right)}^{\eta }{\left(2e^{t/2}-2\right)}^{\zeta }dt$$

is convex of order δ given by

$$\delta =1-\frac{59}{4}\left|\eta \right|-\frac{2}{3}\left|\zeta \right|.$$

Let $n=1,{\xi }_1=3$ and $\varrho =1$ in the Theorem 3, then we get following result.

Corollary 4. 

Let η and ζ be complex numbers such that

$$0<\frac{59}{2}\left|\eta \right|+\frac{2}{3}\left|\zeta \right|\leqq 1.$$

Then the function

$$6^{\eta }{3}^{\zeta }{\int }_0^{\rho }{t}^{-3\eta -3\zeta }{\left(t{e}^t-2e^t+t+2\right)}^{\eta }{\left(2e^t-t^2-2t-2\right)}^{\zeta }dt$$

is convex of order δ given by

$$\delta =1-\frac{59}{2}\left|\eta \right|-\frac{2}{3}\left|\zeta \right|.$$

4. Univalence and Convexity Conditions for the Integral Operator in (6)

In this section, we investigate the univalence and convexity properties for the integral operator defined by (6).

Theorem 4. 

Let $v=1,2,\dots ,n,$${\xi }_v>-1,$$\varrho >0$ and $2\varrho -1<{\xi }_v.$ Also, let $\beta ,$${\eta }_v$ and ${\zeta }_v$ be in $\mathbb{C}$ such that

$$\left|\beta \right|\le 1\left(\beta \ne -1\right)\mathrm{and}\Re \left(1+\sum _{v=1}^n{\eta }_v\right)>0.$$

Assume that these numbers satisfy the following inequality:

$$\left|\beta \right|+\frac{1}{\left|1+{\sum }_{v=1}^n{\eta }_v\right|}\sum _{v=1}^n\left[\left|{\eta }_v\right|\frac{{\left(\xi +1\right)}^2}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}+\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right]\leqq 1,$$

where $\xi =min\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_n\right\}.$ Then the function ${\mathcal{K}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n}^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }$ defined by (6) is in the class $\mathcal{S}.$

Proof. 

Let us define the functions $\psi$ by

$$\psi \left(\rho \right)={\int }_0^{\rho }\prod _{v=1}^n{\left({\mathbb{E}}_{{\xi }_v,{\varrho }_v}\left(t\right)\right)}^{{\eta }_v}{\left(e^{{\mathbb{E}}_{{\xi }_v,{\varrho }_v}\left(t\right)}\right)}^{{\zeta }_v}dt.$$

(25)

Then $\psi \left(0\right)={\psi }^{\prime }\left(0\right)-1=0.$ Differentiating both sides of (25) logarithmically, we get

$$\frac{\rho {\psi }^{″}\left(\rho \right)}{{\psi }^{\prime }\left(\rho \right)}=\sum _{v=1}^n{\eta }_v\left(\frac{\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)}{{\mathbb{E}}_{{\xi }_v,\varrho }\left(\rho \right)}\right)+\sum _{v=1}^n{\zeta }_v\left(\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)\right)$$

and, from (11) and (12) in Lemma 3, we obtain

$$\begin{array}{ccc}\hfill \left|\frac{\rho {\psi }^{″}\left(\rho \right)}{{\psi }^{\prime }\left(\rho \right)}\right|& \leqq & \sum _{v=1}^n\left(\left|{\eta }_v\right|\left|\frac{\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)}{{\mathbb{E}}_{{\xi }_v,\varrho }\left(\rho \right)}\right|+\left|{\zeta }_v\right|\left|\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)\right|\right)\hfill \\ & \leqq & \sum _{v=1}^n\left[\left|{\eta }_v\right|\frac{{\left({\xi }_v+1\right)}^2}{\left({\xi }_v-\varrho +1\right)\left({\xi }_v-2\varrho +1\right)}+\left|{\zeta }_v\right|{\left(\frac{{\xi }_v+1}{{\xi }_v-\varrho +1}\right)}^2\right]\hfill \\ & \leqq & \sum _{v=1}^n\left[\left|{\eta }_v\right|\frac{{\left(\xi +1\right)}^2}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}+\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right].\hfill \end{array}$$

(26)

$$\left(\rho \in \mathbb{D};\xi ,{\xi }_v>2\varrho -1,\left(v=1,2,\dots ,n\right)\right).$$

Here, since the functions

$${\mathsf{\Theta }}_3,{\mathsf{\Theta }}_4:\left(2\varrho -1,\infty \right)⟶\mathbb{R},$$

defined by

$${\mathsf{\Theta }}_3\left(x\right)=\frac{{\left(x+1\right)}^2}{\left(x-\varrho +1\right)\left(x-2\varrho +1\right)}\mathrm{and}{\mathsf{\Theta }}_4\left(x\right)={\left(\frac{x+1}{x-\varrho +1}\right)}^2,$$

are decreasing, the inequalities

$$\frac{{\left({\xi }_v+1\right)}^2}{\left({\xi }_v-\varrho +1\right)\left({\xi }_v-2\varrho +1\right)}<\frac{{\left(\xi +1\right)}^2}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}$$

and

$${\left(\frac{{\xi }_v+1}{{\xi }_v-\varrho +1}\right)}^2<{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2$$

(27)

holds. Thus, we have

$$\begin{array}{ccc}& & \left|\beta {\left|\rho \right|}^{2\left(1+{\sum }_{v=1}^n{\eta }_v\right)}+\left(1-{\left|\rho \right|}^{2\left(1+{\sum }_{v=1}^n{\eta }_v\right)}\right)\frac{\rho {\psi }^{″}\left(\rho \right)}{\left(1+{\sum }_{v=1}^n{\eta }_v\right){\psi }^{\prime }\left(\rho \right)}\right|\hfill \\ & & \le \left|\beta \right|+\left|\frac{\rho {\psi }^{″}\left(\rho \right)}{\left(1+{\sum }_{v=1}^n{\eta }_v\right){\psi }^{\prime }\left(\rho \right)}\right|\hfill \\ & & \le \left|\beta \right|+\frac{1}{\left|1+{\sum }_{v=1}^n{\eta }_v\right|}\sum _{v=1}^n\left[\left|{\eta }_v\right|\frac{{\left(\xi +1\right)}^2}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}+\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right]\hfill \\ & & \le 1.\hfill \end{array}$$

(28)

Using Lemma 1 with

$$\alpha =1+\sum _{v=1}^n{\eta }_v,$$

the inequality (28) imply that the function ${\mathcal{K}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n}^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }\in \mathcal{S}.$ □

Theorem 5. 

Let the parameters $\varrho ,$${\eta }_v,$${\xi }_v$ and ${\zeta }_v$$(v=1,2,\dots ,n)$ be as in Theorem 4. Suppose that $\xi =min\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_n\right\}$ and that the following inequality holds true:

$$\sum _{v=1}^n\left[\left|{\eta }_v\right|\frac{{\left(\xi +1\right)}^2}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}+\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right]\leqq 1.$$

Then the function ${\mathcal{K}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n}^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }$ defined by (6) is in the normalized univalent function class $\mathcal{S}.$

Proof. 

Let us consider the function $\psi$ as in (25). Therefore, from (26) and hypothesis of theorem we can easily see that

$$\begin{array}{ccc}& & \left(1-{\left|\rho \right|}^2\right)\left|\frac{\rho {\psi }^{″}\left(\rho \right)}{{\psi }^{\prime }\left(\rho \right)}\right|\hfill \\ & & \leqq \sum _{v=1}^n\left[\left|{\eta }_v\right|\frac{{\left(\xi +1\right)}^2}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}+\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right]\hfill \\ & & \le 1.\hfill \end{array}$$

(29)

By Lemma 2, with $\varpi =1$ and $\alpha =1+{\sum }_{v=1}^n{\eta }_v,$ the inequality (29) imply that the function ${\mathcal{K}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n}^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }\in \mathcal{S}.$ □

Theorem 6. 

Let $v=1,2,\dots ,n,$${\xi }_v>-1,$$\varrho >0$ and $2\varrho -1<{\xi }_v.$ Also, let ${\eta }_v$ and ${\zeta }_v$ be in $\mathbb{C}$ such that

$$\Re \left(1+\sum _{v=1}^n{\eta }_v\right)>0.$$

Moreover, suppose that the following inequality holds true:

$$0<\sum _{v=1}^n\left[\left|{\eta }_v\right|\frac{{\left(\xi +1\right)}^2}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}+\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right]\leqq 1,$$

where $\xi =min\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_n\right\}.$ Then the function ${\mathcal{K}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n}^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }$ defined by (6), is convex of order δ given by

$$\delta =1-\sum _{v=1}^n\left[\left|{\eta }_v\right|\frac{{\left(\xi +1\right)}^2}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}+\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right].$$

Proof. 

By using (26) we conclude that

$$\begin{array}{ccc}\hfill \left|\frac{\rho {\psi }^{″}\left(\rho \right)}{{\psi }^{\prime }\left(\rho \right)}\right|& \leqq & \sum _{v=1}^n\left[\left|{\eta }_v\right|\left|\frac{\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)}{{\mathbb{E}}_{{\xi }_v,\varrho }\left(\rho \right)}\right|+\left|{\zeta }_v\right|\left|z{\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)\right|\right]\hfill \\ & \leqq & \sum _{v=1}^n\left[\left|{\eta }_v\right|\frac{{\left(\xi +1\right)}^2}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}+\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right]\hfill \\ & =& 1-\delta .\hfill \end{array}$$

This show that, the function $\psi$ is convex of order

$$\delta =1-\sum _{v=1}^n\left[\left|{\eta }_v\right|\frac{{\left(\xi +1\right)}^2}{\left(\xi -\varrho +1\right)\left(\xi -2\varrho +1\right)}+\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right].$$

□

From Theorem 4 with $n=1,{\xi }_1=2$ and $\varrho =1/2,$ we can get the following result.

Corollary 5. 

Let $\beta ,$η and ζ be in $\mathbb{C}$ such that $\Re \left(1+\eta \right)>0$ and $\left|\beta \right|\le 1\left(\beta \ne -1\right).$ If these numbers satisfy the inequality:

$$\left|\beta \right|+\frac{1}{\left|1+\eta \right|}\left(\frac{9}{5}\left|\eta \right|+\frac{36}{25}\left|\zeta \right|\right)\leqq 1,$$

then the function

$${\left[2^{\eta }\left(1+\eta \right){\int }_0^{\rho }{\left(\frac{4\left(2e^{t/2}-t-2\right)}{t}\right)}^{\eta }{e}^{\frac{4\zeta \left(2e^{t/2}-t-2\right)}{t}}dt\right]}^{1/\left(1+\eta \right)}$$

(30)

is in the class $\mathcal{S}.$

Example 2. 

From Corollary 5 with $\eta =1$ and $\zeta =1/8,$ we have

$${\hslash }_2\left(\rho \right)=4{\left[{\int }_0^{\rho }\left(\frac{2e^{t/2}-t-2}{t}\right)e^{\frac{2e^{t/2}-t-2}{2t}}dt\right]}^{1/2}\in \mathcal{S}.$$

In reality, by a simple calculation, we get

$$1+\frac{\rho {\hslash }_2^{″}\left(\rho \right)}{{\hslash }_2^{\prime }\left(\rho \right)}+\frac{\rho {\hslash }_2^{\prime }\left(\rho \right)}{{\hslash }_2\left(\rho \right)}=1+\frac{2e^{\rho }\left(\rho -2\right)+e^{\rho /2}\left({\rho }^2-4\rho +8\right)+2\rho -4}{2\rho \left(2e^{\rho /2}-\rho -2\right)}=g\left(\rho \right).$$

It also holds true that $\Re \left(g\left(\rho \right)\right)>0$ for all $\rho \in \mathbb{D}$ (see Figure 2). Therefore, ${\hslash }_2$ is a $1/2—$convex function [[35], Vol. I, p. 142]. Thus it follows from [[35], Vol. I, p. 142] that ${\hslash }_2$ belongs to the class $\mathcal{S}.$

Figure 2. Image of $\mathbb{D}$ under g.

From Theorem 4 with $n=1,{\xi }_1=3$ and $\varrho =1,$ we can get result below.

Corollary 6. 

Let $\beta ,$η and ζ be in $\mathbb{C}$ such that $\Re \left(1+\eta \right)>0$ and $\left|\beta \right|\le 1\left(\beta \ne -1\right).$ If these numbers satisfy the follwing inequality

$$\left|\beta \right|+\frac{1}{\left|1+\eta \right|}\left(\frac{8}{3}\left|\eta \right|+\frac{16}{9}\left|\zeta \right|\right)\leqq 1,$$

then the function

$${\left[3^{\eta }\left(1+\eta \right){\int }_0^{\rho }{\left(\frac{2e^t-t^2-2t-2}{t^2}\right)}^{\eta }{e}^{\zeta \left(\frac{6e^t-3t^2-6t-6}{t^2}\right)}dt\right]}^{1/\left(1+\eta \right)}$$

(31)

is in the class $\mathcal{S}.$

From Theorem 6 with $n=1,{\xi }_1=2$ and $\varrho =1/2,$ we can get the following result.

Corollary 7. 

Let η and ζ be a complex numbers such that

$$\Re \left(1+\eta \right)>0\mathrm{and}0<\frac{9}{5}\left|\eta \right|+\frac{36}{25}\left|\zeta \right|\leqq 1.$$

Then the function defined by (30) is convex of order δ given by

$$\delta =1-\frac{9}{5}\left|\eta \right|-\frac{36}{25}\left|\zeta \right|.$$

From Theorem 6 with $n=1,{\xi }_1=3$ and $\varrho =1,$ we can get the following result.

Corollary 8. 

Let η and ζ be a complex numbers such that

$$\Re \left(1+\eta \right)>0\mathrm{and}0<\frac{8}{3}\left|\eta \right|+\frac{16}{9}\left|\zeta \right|\leqq 1.$$

Then the function defined by (31) is convex of order δ given by

$$\delta =1-\frac{8}{3}\left|\eta \right|-\frac{16}{9}\left|\zeta \right|.$$

5. Univalence and Convexity Conditions for the Integral Operator in (7)

In this section, we derive the univalence and convexity results for the integral operator defined by (7).

Theorem 7. 

Let $v=1,2,\dots ,n,$${\xi }_v>-1,$$\varrho >0$ and $\left(2+\sqrt{2}\right)\varrho -1<{\xi }_v.$ Also, let $\gamma , \beta , {\eta }_v$ and ${\zeta }_v$ be in $\mathbb{C}$ such that

$$\Re \left(\gamma \right)>0,\left|\beta \right|\le 1\left(\beta \ne -1\right).$$

Assume that these numbers satisfy the following inequality:

$$\left|\beta \right|+\frac{1}{\left|\gamma \right|}\left\{\sum _{v=1}^n\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\sum _{v=1}^n\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right\}\leqq 1,$$

where $\xi =min\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_n\right\}.$ Then the function ${\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }$ defined by (7) is in the class $\mathcal{S}.$

Proof. 

Let us define the function $\varphi$ by

$$\varphi \left(\rho \right):={\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;1}^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }\left(\rho \right)={\int }_0^{\rho }\prod _{v=1}^n{\left({\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(t\right)\right)}^{{\eta }_v}{\left(e^{{\mathbb{E}}_{{\xi }_v,\varrho }\left(t\right)}\right)}^{{\zeta }_v}dt,$$

(32)

so that, obviously,

$${\varphi }^{\prime }\left(\rho \right)=\prod _{v=1}^n{\left({\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)\right)}^{{\eta }_v}{\left(e^{{\mathbb{E}}_{{\xi }_v,\varrho }\left(\rho \right)}\right)}^{{\zeta }_v}$$

(33)

and $\varphi \left(\rho \right)={\varphi }^{\prime }\left(\rho \right)-1=0.$

Now we differentiate (33) logarithmically and multiply by $\rho$, we obtain

$$\frac{\rho {\varphi }^{″}\left(\rho \right)}{{\varphi }^{\prime }\left(\rho \right)}=\sum _{v=1}^n{\eta }_v\frac{\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{″}\left(\rho \right)}{{\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)}+\sum _{v=1}^n{\zeta }_v\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right).$$

Furthermore, by (11), (13), (23) and (27) we obtain

$$\begin{array}{ccc}\hfill \left|\frac{\rho {\varphi }^{″}\left(\rho \right)}{{\varphi }^{\prime }\left(\rho \right)}\right|& \leqq & \sum _{v=1}^n\left(\left|{\eta }_v\right|\left|\frac{\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{″}\left(\rho \right)}{{\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)}\right|+\left|{\zeta }_v\right|\left|\rho {\mathbb{E}}_{{\xi }_v,\varrho }^{\prime }\left(\rho \right)\right|\right)\hfill \\ & \leqq & \sum _{v=1}^n\left[\left|{\eta }_v\right|\frac{{\left({\xi }_v-\varrho +1\right)}^3+2\varrho {\left({\xi }_v+1\right)}^2}{{\left({\xi }_v-\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2{\xi }_v\varrho +1\right)}+\left|{\zeta }_v\right|{\left(\frac{{\xi }_v+1}{{\xi }_v-\varrho +1}\right)}^2\right]\hfill \\ & \leqq & \sum _{v=1}^n\left[\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right].\hfill \end{array}$$

(34)

Hence, from (34) we have

$$\begin{array}{ccc}& & \left|\beta {\left|\rho \right|}^{2\gamma }+\left(1-{\left|\rho \right|}^{2\gamma }\right)\frac{\rho {\varphi }^{″}\left(\rho \right)}{\gamma {\varphi }^{\prime }\left(\rho \right)}\right|\hfill \\ & & \leqq \left|\beta \right|+\left|\frac{\rho {\varphi }^{″}\left(\rho \right)}{{\varphi }^{\prime }\left(\rho \right)}\right|\hfill \\ & & \leqq \left|\beta \right|+\frac{1}{\left|\gamma \right|}\left\{\sum _{v=1}^n\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\sum _{v=1}^n\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right\}\hfill \\ & & \leqq 1,\hfill \end{array}$$

which, in view of Lemma 1, implies that ${\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }\in \mathcal{S}.$ □

Theorem 8. 

Let the parameters $\varrho ,$$\gamma , {\eta }_v,$${\xi }_v$ and ${\zeta }_v$$(v=1,2,\dots ,n)$ be as in Theorem 7. Suppose that $\xi =min\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_n\right\}$ and that the following inequality holds true:

$$\Re \left(\gamma \right)\geqq \left\{\sum _{v=1}^n\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\sum _{v=1}^n\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right\}.$$

Then the function ${\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }$ defined by (7) is in the class $\mathcal{S}.$

Proof. 

By using (34) we obtain

$$\begin{array}{ccc}& & \frac{1-{\left|\rho \right|}^{2\Re \left(\gamma \right)}}{\Re \left(\gamma \right)}\left|\frac{\rho {\varphi }^{″}\left(\rho \right)}{{\varphi }^{\prime }\left(\rho \right)}\right|\hfill \\ & & \leqq \frac{1-{\left|\rho \right|}^{2\Re \left(\gamma \right)}}{\Re \left(\gamma \right)}\left\{\sum _{v=1}^n\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\sum _{v=1}^n\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right\}\hfill \\ & & \leqq \frac{1}{\Re \left(\gamma \right)}\left\{\sum _{v=1}^n\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\sum _{v=1}^n\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\right\}\hfill \\ & & \leqq 1,\hfill \end{array}$$

which, in view of Lemma 2, implies that ${\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;\gamma }^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho }\in \mathcal{S}.$ □

Theorem 9. 

Let $v=1,2,\dots ,n,$${\xi }_v>-1,$$\varrho >0$ and $\left(2+\sqrt{2}\right)\varrho -1<{\xi }_v.$ Also, let $\gamma ,{\eta }_v$ and ${\zeta }_v$ be in $\mathbb{C}$ such that $\Re \left(\gamma \right)>0.$ Assume that these numbers satisfy the following inequality:

$$0<\sum _{v=1}^n\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\sum _{v=1}^n\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\leqq 1,$$

where $\xi =min\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_n\right\}.$ Then the function ${\mathcal{L}}_{{\eta }_1,{\eta }_2,\dots ,{\eta }_n;{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n;n;1}^{{\xi }_1,{\xi }_2,\dots ,{\xi }_n;\varrho },$ defined by (7) with $\gamma =1,$ is convex of order δ given by

$$\delta =1-\sum _{v=1}^n\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}-\sum _{v=1}^n\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2.$$

Proof. 

From (34) and hypothesis of theorem, we obtain

$$\begin{array}{ccc}\hfill \left|\frac{\rho {\varphi }^{″}\left(\rho \right)}{{\varphi }^{\prime }\left(\rho \right)}\right|& \leqq & \sum _{v=1}^n\left|{\eta }_v\right|\frac{{\left(\xi -\varrho +1\right)}^3+2\varrho {\left(\xi +1\right)}^2}{{\left(\xi -\varrho +1\right)}^2+{\left(\varrho -1\right)}^2-\left(2\xi \varrho +1\right)}+\sum _{v=1}^n\left|{\zeta }_v\right|{\left(\frac{\xi +1}{\xi -\varrho +1}\right)}^2\hfill \\ & =& 1-\delta .\hfill \end{array}$$

Therefore, the function $\varphi$ is convex of order $\delta .$ □

From Theorem 7 with $n=1,{\xi }_1=1$ and $\varrho =1/2,$ we can get following result.

Corollary 9. 

Let $\gamma ,\beta ,\eta$ and ζ be in $\mathbb{C}$ such that $\Re \left(\gamma \right)>0,\left|\beta \right|\le 1\left(\beta \ne -1\right).$ If these numbers satisfy the inequality:

$$\left|\beta \right|+\frac{1}{\left|\gamma \right|}\left(\frac{59}{4}\left|\eta \right|+\frac{16}{9}\left|\zeta \right|\right)\leqq 1$$

then the function

$${\left[\gamma {\int }_0^{\rho }{t}^{\gamma -1}{\left(e^{t/2}\right)}^{\eta }{\left(e^{2e^{t/2}-2}\right)}^{\zeta }dt\right]}^{1/\gamma }$$

is in the normalized univalent function class $\mathcal{S}.$

Example 3. 

From Corollary 9 with $\beta =0,\gamma =1,\eta =0.01$ and $\zeta =0.1,$ we obtain

$${\hslash }_3\left(\rho \right)={\int }_0^{\rho }\left(e^{t/200}\right)\left(e^{\frac{e^{t/2}-1}{5}}\right)dt\in \mathcal{S}.$$

From Theorem 7 with $n=1,{\xi }_1=3$ and $\varrho =1,$ we can get result below.

Corollary 10. 

Let $\gamma ,\beta ,\eta$ and ζ be in $\mathbb{C}$ such that $\Re \left(\gamma \right)>0$ and $\beta \ne -1.$ If these numbers satisfy the inequality

$$\left|\beta \right|+\frac{1}{\left|\gamma \right|}\left(\frac{59}{2}\left|\eta \right|+\frac{16}{9}\left|\zeta \right|\right)\leqq 1$$

then the function

$${\left[6^{\eta }\gamma {\int }_0^{\rho }{t}^{-3\eta +\gamma -1}{\left(t{e}^t-2e^t+t+2\right)}^{\eta }{e}^{3\zeta \left(\frac{2e^t-t^2-2t-2}{t^2}\right)}dt\right]}^{1/\gamma }$$

is in the normalized univalent function class $\mathcal{S}.$

From Theorem 9 with $n=1,{\xi }_1=1$ and $\varrho =1/2,$ we have following result.

Corollary 11. 

Let η and ζ be complex numbers such that

$$\frac{59}{4}\left|\eta \right|+\frac{16}{9}\left|\zeta \right|\leqq 1.$$

Then the function

$${\int }_0^{\rho }{\left(e^{t/2}\right)}^{\eta }{\left(e^{2e^{t/2}-2}\right)}^{\zeta }dt$$

is convex of order δ given by

$$\delta =1-\frac{59}{4}\left|\eta \right|-\frac{16}{9}\left|\zeta \right|.$$

From Theorem 9 with $n=1,{\xi }_1=3$ and $\varrho =1,$ we have following result.

Corollary 12. 

Let η and ζ be complex numbers such that

$$\frac{59}{2}\left|\eta \right|+\frac{16}{9}\left|\zeta \right|\leqq 1.$$

Then the function

$$6^{\eta }{\int }_0^{\rho }{t}^{-3\eta }{\left(t{e}^t-2e^t+t+2\right)}^{\eta }{e}^{3\zeta \left(\frac{2e^t-t^2-2t-2}{t^2}\right)}dt$$

is convex of order δ given by

$$\delta =1-\frac{59}{2}\left|\eta \right|-\frac{16}{9}\left|\zeta \right|.$$

Example 4. 

From Corollary 12 with $\eta =0.01$ and $\zeta =0.1,$ we get

$${\hslash }_4\left(\rho \right)=6^{1/100}{\int }_0^{\rho }{t}^{-3/100}{\left(t{e}^t-2e^t+t+2\right)}^{1/100}{e}^{3\left(\frac{2e^t-t^2-2t-2}{10t^2}\right)}dt$$

is convex of order $\delta =949/1800.$

6. Conclusions

In the present investigation, we first introduced certain families of integral operators by using the Miller–Ross function which, in particular, plays a very important role in the study of pure and applied mathematical sciences. Therefore, it is important to know the geometric properties of special functions and their integral operators. For this reason, we aim to study the criteria for the univalence and convexity of these integral operators that are defined by using Miller–Ross functions. The various results, which we established in this paper, are believed to be new, and their importance is illustrated by several interesting consequences and examples together with the associated graphical illustrations.

Hopefully, the original results contained here would stimulate researchers’ imagination and inspire them, just as all the operators introduced before in studies related to functions of a complex variable have done. Other geometric properties related to them could be investigated, and also they could prove useful in introducing special classes of functions based on those properties.