Geometric Properties of Normalized Mittag–Leffler Functions

Abstract

The aim of this paper is to investigate certain properties such as convexity of order $\mu$, close-to-convexity of order $\left(1+\mu \right)$/2 and starlikeness of normalized Mittag–Leffler function. We use some inequalities to prove our results. We also discuss the close-to-convexity of Mittag–Leffler functions with respect to certain starlike functions. Furthermore, we find the conditions for the above-mentioned function to belong to the Hardy space ${\mathcal{H}}^p$. Some of our results improve the results in the literature.

1. Introduction

The one parameter Mittag–Leffler function ${\mathbb{E}}_{\alpha }\left(z\right)$ defined by

$${\mathbb{E}}_{\alpha }\left(z\right)=\stackrel{\infty }{\sum _{m=0}}\frac{z^m}{\Gamma \left(\alpha m+1\right)}$$

(1)

was introduced by Mittag–Leffler [1]. This function of complex variable is entire. The series defined by Equation (1) converges in $\mathbb{C}$ when $Re\left(\alpha \right)>0.$ Consider that the function ${\mathbb{E}}_{\alpha ,\kappa }\left(z\right)$ which generalizes the function ${\mathbb{E}}_{\alpha }\left(z\right)$ is defined by

$${\mathbb{E}}_{\alpha ,\kappa }\left(z\right)=\stackrel{\infty }{\sum _{m=0}}\frac{z^m}{\Gamma \left(\alpha m+\kappa \right)},\alpha ,\kappa \in \mathbb{C},z\in \mathbb{C}.$$

(2)

It was introduced by Wiman [2] and was named as Mittag–Leffler type function. The series in Equation (2) converges in $\mathbb{C}$ when $Re\left(\alpha \right)>0$ and $Re\left(\kappa \right)>0$. Furthermore, the functions defined in (1) and (2) are entire of order $1/Re\left(\alpha \right)$ and of type $1,$ for more details, see [3]. The function ${\mathbb{E}}_{\alpha ,\kappa }\left(z\right)$ and its analysis with its generalizations is increasingly becoming a rich research area in mathematics and its related fields. A number of researchers studied and analyzed the function given in (2) (see Wiman [2,4,5]). One can find this function in the study of kinetic equation of fractional order, Lévy flights, random walks, super-diffusive transport as well as in investigations of complex systems. In a similar manner, the advanced studies of these functions reflect and highlight many vital properties of these functions. The function ${\mathbb{E}}_{\alpha ,\kappa }\left(z\right)$ generalizes many functions such as

$$\begin{array}{ccc}\hfill {\mathbb{E}}_{1,1}\left(z\right)& =& e^z,{\mathbb{E}}_{1,2}\left(z\right)=\frac{e^z-1}{z},\hfill \\ \hfill {\mathbb{E}}_{2,1}\left(z\right)& =& cosh\left(\sqrt{z}\right),{\mathbb{E}}_{2,2}\left(z\right)=\frac{sinh\left(\sqrt{z}\right)}{\sqrt{z}}.\hfill \end{array}$$

The interested readers are suggested to go through [6,7,8,9].

Let $\mathcal{A}$ be the family of all functions g having the form

$$g\left(z\right)=z+\stackrel{\infty }{\sum _{m=2}}{a}_m{z}^m,$$

(3)

and are analytic in $\mathcal{D}=\left\{z:\left|z\right|<1\right\}$ and $\mathcal{S}$ denote the family of univalent functions from $\mathcal{A}$. The families of functions which are convex, starlike and close-to-convex of order $\mu$, respectively, are defined as:

$$\begin{array}{ccc}\hfill \mathcal{C}\left(\mu \right)& =& \left\{g:g\in \mathcal{A}\mathrm{and}Re\left(1+\frac{z{g}^{″}\left(z\right)}{g^{\prime }\left(z\right)}\right)>\mu ,z\in \mathcal{D};0\le \mu <1\right\},\hfill \\ \hfill {\mathcal{S}}^{\ast }\left(\mu \right)& =& \left\{g:g\in \mathcal{A}\mathrm{and}Re\left(\frac{z{g}^{\prime }\left(z\right)}{g\left(z\right)}\right)>\mu ,z\in \mathcal{D};0\le \mu <1\right\},\hfill \end{array}$$

and

$$\mathcal{K}\left(\mu \right)=\left\{g:g\in \mathcal{A}\mathrm{and}Re\left(\frac{g^{\prime }\left(z\right)}{h^{\prime }\left(z\right)}\right)>\mu ,z\in \mathcal{D};0\le \mu <1;h\in \mathcal{C}\right\}.$$

It is clear that ${\mathcal{C}}^{\ast }\left(0\right)=\mathcal{C}$, ${\mathcal{S}}^{\ast }\left(0\right)={\mathcal{S}}^{\ast }$ and $\mathcal{K}\left(0\right)=\mathcal{K}.$ Consider the class $\mathcal{H}$ of all analytic functions in $\mathcal{D}$ and $\mu <1,$ Baricz [10] introduced the classes

$${\mathcal{P}}_{\eta }\left(\mu \right)=\left\{p:p\in \mathcal{H},p\left(0\right)=1,Re\left\{e^{i\eta }\left(p\left(z\right)-\mu \right)\right\}>0,z\in \mathcal{D},\eta \in \mathbb{R}\right\}$$

and

$${\mathcal{R}}_{\eta }\left(\mu \right)=\left\{f:f\in \mathcal{A}\mathrm{and}Re\left\{e^{i\eta }\left(f^{\prime }\left(z\right)-\mu \right)\right\}>0,z\in \mathcal{D},\eta \in \mathbb{R}\right\}.$$

For $\eta =0,$ we have the classes of analytic functions ${\mathcal{P}}_0\left(\alpha \right)$ and ${\mathcal{R}}_0\left(\alpha \right)$ respectively. Also for $\eta =0$ and $\alpha =0,$ we have the classes $\mathcal{P}$ and $\mathcal{R}$.

For the functions $g\in \mathcal{A}$ given by (1) and $h\in \mathcal{A}$ given by

$$h\left(z\right)=z+\sum _{m=2}^{\infty }{b}_m{z}^m,$$

then the convolution (Hadamard product) of g and h is defined as:

$$\left(g\ast h\right)\left(z\right)=z+\sum _{m=2}^{\infty }{a}_m{b}_m{z}^m,z\in \mathcal{D}.$$

It is clear that the function ${\mathbb{E}}_{\alpha ,\kappa }\left(z\right)$ is not in class $\mathcal{A}$. Recently, Bansal and Prajapat [11] considered the normalization of the function ${\mathbb{E}}_{\alpha ,\kappa }\left(z\right)$ given as

$$E_{\alpha ,\kappa }\left(z\right)=\Gamma \left(\kappa \right)z{\mathbb{E}}_{\alpha ,\kappa }\left(z\right)=z+\stackrel{\infty }{\sum _{m=1}}\frac{\Gamma \left(\kappa \right)z^{m+1}}{\Gamma \left(\alpha m+\kappa \right)},\alpha ,\kappa \in \mathbb{C},Re\left(\alpha \right)>0,\kappa \ne 0,-1,\cdots .$$

In this article, we investigate some geometric properties of function $E_{\alpha ,\kappa }\left(z\right)$ with real parameters $\alpha$ and $\kappa$.

We need the following results in our investigations.

Lemma 1

([12]). If $g\in \mathcal{A}$ and

$$\left|z{g}^{″}\left(z\right)\right|<\frac{1-\mu }{4},z\in \mathcal{D};0\le \mu <1,$$

then

$$Re\left\{g^{\prime }\left(z\right)\right\}>\frac{1+\mu }{2},z\in \mathcal{D};0\le \mu <1.$$

Lemma 2

([13]). Let $\kappa \in \mathbb{C}$ such that $Re\left(\kappa \right)>0,$ $c\in \mathbb{C}$ and $\left|c\right|\le 1,$ $c\ne -1.$ If $h\in \mathcal{A}$ satisfies

$$\left|c{\left|z\right|}^{2\beta }+\left(1-{\left|z\right|}^{2\beta }\frac{z{h}^{″}\left(z\right)}{\beta h^{\prime }\left(z\right)}\right)\right|\le 1,z\in \mathcal{D},$$

then

$$C_{\beta }\left(z\right)={\left\{\beta \underset{0}{\overset{z}{\int }}{t}^{\beta -1}{h}^{\prime }\left(t\right)dt\right\}}^{1/\beta },z\in \mathcal{D}$$

is analytic and univalent in $\mathcal{D}$.

Lemma 3

([14]). Let $g\left(z\right)=z+a_2{z}^2+\dots +a_m{z}^m+\dots$, be analytic in $\mathcal{D}$ and in addition $1\ge 2a_2\ge \dots \ge m{a}_m\ge \dots \ge 0$ or $1\le 2a_2\le \dots \le m{a}_m\le \dots \le 2,$ then $g\left(z\right)$ is in class $\mathcal{K}$ with respect to the function $z\to -log\left(1-z\right).$ Also if the function $g\left(z\right)=z+3a_3+\dots +a_{2m-1}{z}^{2m-1}+\dots ,$ which is odd and analytic in $\mathcal{D}$ and satisfies in addition $1\ge 3a_3\ge \dots \ge \left(2m+1\right)a_{2m+1}\ge \dots \ge 0$ or $1\le 3a_3\le \dots \le \left(2m+1\right)a_{2m+1}\le \dots \le 2,$ then $g\left(z\right)\in \mathcal{S}$ in $\mathcal{D}.$

Lemma 4

([15]). If $g\left(z\right)={\displaystyle \sum _{m=1}^{\infty }}{a}_m{z}^{m-1},$ such that $a_1=1$ and $a_m\ge 0,$ $\forall m\ge 2,$ is analytic in $\mathcal{D}$ and if ${\left\{a_m\right\}}_{m=1}^{\infty }$ is a sequence which is decreasing, i.e., $a_{m+2}+a_m-2a_{m+1}\ge 0$ and $a_m-a_{m+1}\ge 0,$ $\forall m\ge 1,$ then

$$Re\left\{\sum _{m=1}^{\infty }{a}_m{z}^{m-1}\right\}>\frac{1}{2},\forall z\in \mathcal{D}.$$

Lemma 5

([15]). If $a_m\ge 0,$ $\left\{m{a}_m\right\}$ and $\{m{a}_m-(m+1)a_{m+1}\}$ both are non-increasing, then the function g defined by (3) is in ${\mathcal{S}}^{\ast }.$

2. Starlikeness, Convexity, Close-to-Convexity

Theorem 1.

Let$\alpha \ge \frac{3}{2}$and$\kappa \ge \frac{3}{2}.$Then,

$$Re\left\{\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\right\}>\frac{1}{2},forz\in \mathcal{D}.$$

Proof. 

For the proof of this result, we have to show that

$${\left\{a_m\right\}}_{m=1}^{\infty }={\left\{\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha (m-1)+\kappa \right)}\right\}}_{m=1}^{\infty }$$

is a decreasing sequence. Consider

$$\begin{array}{ccc}\hfill a_m-a_{m+1}& =& \frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha (m-1)+\kappa \right)}-\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}\hfill \\ & =& \Gamma \left(\kappa \right)\left\{\frac{\Gamma \left(\alpha m+\kappa \right)-\Gamma \left(\alpha (m-1)+\kappa \right)}{\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha m+\kappa \right)}\right\}>0,\hfill \end{array}$$

where ∀$m\ge 1,$ $\alpha \ge \frac{3}{2}$ and $\kappa \ge \frac{3}{2}.$ Now, to show that ${\left\{a_m\right\}}_{m=1}^{\infty }$ is decreasing, we prove that $a_m+a_{m+2}\ge 2a_{m+1}$.

Take

$$\begin{array}{c}{a}_m-2a_{m+1}+a_{m+2}=\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha (m+1)+\kappa \right)}+\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha (m-1)+\kappa \right)}-\frac{2\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}\\ =\Gamma \left(\kappa \right)\left\{\frac{\begin{array}{c}\Gamma \left(\alpha m+\kappa \right)\Gamma \left(\alpha (m+1)+\kappa \right)-2\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha (m+1)+\kappa \right)\\ +\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha m+\kappa \right)\end{array}}{\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha m+\kappa \right)\Gamma \left(\alpha (m+1)+\kappa \right)}\right\}\\ =\Gamma \left(\kappa \right)\left[\frac{\begin{array}{c}\Gamma \left(\alpha (m+1)+\kappa \right)\left\{\Gamma \left(\alpha m+\kappa \right)-2\Gamma \left(\alpha (m-1)+\kappa \right)\right\}\\ +\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha m+\kappa \right)\end{array}}{\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha m+\kappa \right)\Gamma \left(\alpha (m+1)+\kappa \right)}\right].\end{array}$$

The above expression is non negative ∀$m\ge 1,$ $\alpha \ge \frac{3}{2}$ and $\kappa \ge \frac{3}{2},$ which shows that ${\left\{a_m\right\}}_{m=1}^{\infty }$ is decreasing and convex sequence. Now, from the Lemma 4, we have

$$Re\left(\sum _{m=1}^{\infty }{b}_m{z}^{m-1}\right)>\frac{1}{2},z\in \mathcal{D},$$

which is equivalent to

$$Re\left(\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\right)>\frac{1}{2},z\in \mathcal{D}.$$

 □

Theorem 2.

Let$\alpha \ge 2.67$and$\kappa \ge 1.$Then,$E_{\alpha ,\kappa }\left(z\right)$is starlike in the open unit disc$\mathcal{D}$.

Proof. 

To show that $E_{\alpha ,\kappa }\left(z\right)$ is starlike in $\mathcal{D},$ we prove that $\left\{m{a}_m\right\}$ and $\{m{a}_m-(m+1)a_{m+1}\}$ both are non-increasing in view of Lemma 5. Since $a_m\ge 0$ for the normalized Mittag–Leffler function under the given conditions, consider

$$\begin{array}{ccc}\hfill m{a}_m-(m+1)a_{m+1}& =& \frac{m\Gamma \left(\kappa \right)}{\Gamma \left(\alpha (m-1)+\kappa \right)}-\frac{(m+1)\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}\hfill \\ & =& \Gamma \left(\kappa \right)\left\{\frac{m\Gamma \left(\alpha m+\kappa \right)-(m+1)\Gamma \left(\alpha (m-1)+\kappa \right)}{\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha m+\kappa \right)}\right\}>0\hfill \end{array}$$

for $m\ge 1,$ $\alpha \ge 2.67$ and $\kappa \ge 1.$ Now,

$$\begin{array}{c}m{a}_m-2(m+1)a_{m+1}+(m+2)=\frac{m\Gamma \left(\kappa \right)}{\Gamma \left(\alpha (m-1)+\kappa \right)}-\frac{2(m+1)\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}+\frac{(m+2)\Gamma \left(\kappa \right)}{\Gamma \left(\alpha (m+1)+\kappa \right)}\\ =\Gamma \left(\kappa \right)\left\{\frac{\begin{array}{c}-2(m+1)\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha (m+1)+\kappa \right)+\\ m\Gamma \left(\alpha m+\kappa \right)\Gamma \left(\alpha (m+1)+\kappa \right)+(m+2)\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha m+\kappa \right)\end{array}}{\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha m+\kappa \right)\Gamma \left(\alpha (m+1)+\kappa \right)}\right\}\\ =\Gamma \left(\kappa \right)\left[\frac{\begin{array}{c}\Gamma \left(\alpha (m+1)+\kappa \right)\left\{m\Gamma \left(\alpha m+\kappa \right)-2(m+1)\Gamma \left(\alpha (m-1)+\kappa \right)\right\}\\ +(m+2)\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha m+\kappa \right)\end{array}}{\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha m+\kappa \right)\Gamma \left(\alpha (m+1)+\kappa \right)}\right].\end{array}$$

The above relation is non-negative ∀$m\ge 1,$ $\alpha \ge 2.67$ and $\kappa \ge 1.$ Thus, from Lemma 5, $E_{\alpha ,\kappa }\left(z\right)$ is starlike in $\mathcal{D}$. □

Theorem 3.

Let$\alpha \ge 3.323$and$\kappa \ge 1.$Then,

$$Re\left\{E_{\alpha ,\kappa }^{\prime }\left(z\right)\right\}>\frac{1}{2},\left(z\in \mathcal{D}\right).$$

Proof. 

Consider

$$\begin{array}{ccc}\hfill E_{\alpha ,\kappa }\left(z\right)& =& z+\stackrel{\infty }{\sum _{m=2}}\frac{\Gamma \left(\kappa \right)z^m}{\Gamma \left(\alpha (m-1)+\kappa \right)},\hfill \\ \hfill E_{\alpha ,\kappa }^{\prime }\left(z\right)& =& 1+\stackrel{\infty }{\sum _{m=2}}\frac{m\Gamma \left(\kappa \right)}{\Gamma \left(\alpha (m-1)+\kappa \right)}{z}^{m-1},\hfill \\ \hfill E_{\alpha ,\kappa }^{\prime }\left(z\right)& =& 1+\stackrel{\infty }{\sum _{m=2}}{A}_m{z}^{m-1}.\hfill \end{array}$$

Here, $A_m=\frac{m\Gamma \left(\kappa \right)}{\Gamma \left(\alpha (m-1)+\kappa \right)}.$ By taking the same computations as in Theorem 2, we get the proof. □

Theorem 4.

If$\alpha \ge 1$and$\kappa \ge 1,$then$z\to E_{\alpha ,\kappa }\left(z\right)$is in$\mathcal{K}$with respect to the function$-log\left(1-z\right).$

Proof. 

Set

$$E_{\alpha ,\kappa }\left(z\right)=z+\underset{m=2}{\sum ^{\infty }}{a}_{m-1}{z}^m,$$

and we have $a_{m-1}>0$ for all $m\ge 2$ and $a_1=\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha +\kappa \right)}\le 1.$ For the proof of this result, we use Lemma 3. Therefore, we have to show that ${\left\{m{a}_{m-1}\right\}}_{m\ge 2}$ is decreasing. Now,

$$\begin{array}{cc}& m{a}_{m-1}-\left(m+1\right)a_m=\Gamma \left(\kappa \right)\left[\frac{m}{\Gamma \left(\alpha (m-1)+\kappa \right)}-\frac{m+1}{\Gamma \left(\alpha m+\kappa \right)}\right],\hfill \\ =& \Gamma \left(\kappa \right)\left[\frac{m\Gamma \left(\alpha m+\kappa \right)-\left(m+1\right)\Gamma \left(\alpha (m-1)+\kappa \right)}{\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha m+\kappa \right)}\right]>0.\hfill \end{array}$$

By restricting parameters, we note that $m{a}_{m-1}-\left(m+1\right)a_m>0$ for all $m\ge 2.$ Thus, ${\left\{m{a}_{m-1}\right\}}_{m\ge 2}$ is a decreasing sequence—hence the result. □

Theorem 5.

If$\alpha \ge 1$and$\kappa \ge 1,$then$z\to z{E}_{\alpha ,\kappa }\left(z^2\right)$is in$\mathcal{K}$respect to the function$\frac{1}{2}log\left(\frac{1+z}{1-z}\right).$

Proof. 

Set

$$z{E}_{\alpha ,\kappa }\left(z^2\right)=z+\underset{m=2}{\sum ^{\infty }}{A}_{2m-1}{z}^{2m-1}.$$

Here, $A_{2m-1}=a_{m-1}=\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha (m-1)+\kappa \right)}$ for all $m\ge 2.$ In addition, it is clear that $a_1\le 1.$ Mainly, we have to show that ${\left\{\left(2m-1\right)a_{m-1}\right\}}_{m\ge 2}$ is decreasing. Now,

$$\begin{array}{cc}& \left(2m-1\right)a_{m-1}-\left(2m+1\right)a_m=\Gamma \left(\kappa \right)\left[\frac{\left(2m-1\right)}{\Gamma \left(\alpha (m-1)+\kappa \right)}-\frac{\left(2m+1\right)}{\Gamma \left(\alpha m+\kappa \right)}\right],\hfill \\ =& \Gamma \left(\kappa \right)\left[\frac{\left(2m-1\right)\Gamma \left(\alpha m+\kappa \right)-\left(2m+1\right)\Gamma \left(\alpha (m-1)+\kappa \right)}{\Gamma \left(\alpha (m-1)+\kappa \right)\Gamma \left(\alpha m+\kappa \right)}\right]>0.\hfill \end{array}$$

By using conditions on parameters, we observe that $\left(2m-1\right)a_{m-1}-\left(2m+1\right)a_m>0$ for all $m\ge 2.$ Thus, ${\left\{\left(2m-1\right)a_{m-1}\right\}}_{m\ge 2}$ is a decreasing sequence. By applying Lemma 3, we have the required result. □

Theorem 6.

If$\alpha \ge 1$and$\kappa \ge 3.214319744,$ then $E_{\alpha ,\kappa }\left(z\right)\in {\mathcal{S}}^{\ast }$ in $\mathcal{D}$.

Proof. 

Let $p\left(z\right)=\frac{z{E}_{\alpha ,\kappa }^{\prime }\left(z\right)}{E_{\alpha ,\kappa }\left(z\right)},$ $z\in \mathcal{D}.$ Then, the function p is analytic in $\mathcal{D}$ with $p\left(0\right)=1.$ To prove $E_{\alpha ,\kappa }\left(z\right)$ is starlike in $\mathcal{D}$, we just prove that $Rep\left(z\right)>0$ in $z\in \mathcal{D}.$ For this, it is enough to show $\left|p\left(z\right)-1\right|<1$ for $z\in \mathcal{D}.$ By using the inequalities

$$\begin{array}{ccc}\hfill \frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}& \le & \frac{1}{{\left(\kappa \right)}_m},\alpha \ge 1,\kappa \ge 1,m\in \mathbb{N},\hfill \\ \hfill m{\left(\kappa \right)}_m& \le & 2^{m-1}\kappa {\left(\kappa +1\right)}_{m-1},\kappa \ge 1,m\in \mathbb{N},\hfill \end{array}$$

we have

$$\begin{array}{ccc}\hfill \left|E_{\alpha ,\kappa }^{\prime }\left(z\right)-\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\right|& =& \left|\underset{m=1}{\sum ^{\infty }}\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}m{z}^m\right|\hfill \\ & \le & \underset{m=1}{\sum ^{\infty }}\frac{2^{m-1}}{\kappa {\left(\kappa +1\right)}^{m-1}}\hfill \\ & \le & \frac{1}{\kappa }\underset{m=1}{\sum ^{\infty }}{\left(\frac{2}{\kappa +1}\right)}^{m-1}\hfill \\ & =& \frac{\kappa +1}{\kappa \left(\kappa -1\right)},\left(\kappa >1\right).\hfill \end{array}$$

(4)

Furthermore, using reverse triangle inequality and the inequality ${\left(\kappa \right)}^m\le {\left(\kappa \right)}_m,$ we obtain

$$\begin{array}{ccc}\hfill \left|\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\right|& =& \left|1+\underset{m=1}{\sum ^{\infty }}\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}{z}^m\right|\hfill \\ & \ge & 1-\underset{m=1}{\sum ^{\infty }}\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}\hfill \\ & \ge & 1-\underset{m=1}{\sum ^{\infty }}\frac{1}{{\left(\kappa \right)}_m}\hfill \\ & \ge & 1-\frac{1}{\kappa }\underset{m=1}{\sum ^{\infty }}{\left(\frac{1}{\kappa +1}\right)}^{m-1}\hfill \\ & =& \frac{{\kappa }^2-\kappa -1}{{\kappa }^2}\left(\kappa >0\right).\hfill \end{array}$$

(5)

By combining (4) and (5), we get

$$\left|\frac{z{E}_{\alpha ,\kappa }^{\prime }\left(z\right)}{E_{\alpha ,\kappa }\left(z\right)}-1\right|\le \frac{\kappa \left(\kappa +1\right)}{\left(\kappa -1\right)\left({\kappa }^2-\kappa -1\right)}.$$

(6)

Therefore, $E_{\alpha ,\kappa }\left(z\right)\in {\mathcal{S}}^{\ast }$ in $\mathcal{D}$ if $\frac{\kappa \left(\kappa +1\right)}{\left(\kappa -1\right)\left({\kappa }^2-\kappa -1\right)}\le 1$. In other words, we have to show that ${\kappa }^3-3{\kappa }^2-\kappa +1\ge 0$. The inequality is satisfied for $\kappa \ge 3.214319744$. Hence, $E_{\alpha ,\kappa }\left(z\right)$ is starlike in $\mathcal{D}$. □

Remark 1.

Recently, Bansal and Prajpat [11] proved that$E_{\alpha ,\kappa }\left(z\right)$is starlike, if$\alpha \ge 1$and$\kappa \ge (3+\sqrt{17})/2\approx 3.56155281.$The above result improves the result in [11].

Theorem 7.

If$\alpha \ge 1$and$\kappa \ge 3.56155281,$then$E_{\alpha ,\kappa }\left(z\right)\in \mathcal{C}$in$\mathcal{D}$.

Proof. 

Let $p\left(z\right)=1+\frac{z{E}_{\alpha ,\kappa }^{″}\left(z\right)}{E_{\alpha ,\kappa }^{\prime }\left(z\right)},$ $z\in \mathcal{D}.$ Then, $p\left(z\right)$ is analytic in $\mathcal{D}$ with $p\left(0\right)=1.$ To show that $E_{\alpha ,\kappa }\left(z\right)$ is convex in $\mathcal{D}$, it is enough to prove that $\left|p\left(z\right)-1\right|<1,$ $z\in \mathcal{D}.$ By using the inequalities

$$\begin{array}{ccc}\hfill \frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}& \le & \frac{1}{{\left(\kappa \right)}_m},\alpha \ge 1,\kappa \ge 1,m\in \mathbb{N},\hfill \\ \hfill 2m\left(m+1\right){\left(\kappa \right)}_m& \le & 4^{m-1}\kappa {\left(\kappa +1\right)}_{m-1},\kappa \ge 1,m\in \mathbb{N},\hfill \end{array}$$

we have

$$\begin{array}{ccc}\hfill \left|z{E}_{\alpha ,\kappa }^{″}\left(z\right)\right|& =& \left|\underset{m=1}{\sum ^{\infty }}\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}m\left(m+1\right)z^m\right|\hfill \\ & \le & \underset{m=1}{\sum ^{\infty }}\frac{4^{m-1}}{2\kappa {\left(\kappa +1\right)}^{m-1}}\hfill \\ & \le & \frac{2}{\kappa }\underset{m=1}{\sum ^{\infty }}{\left(\frac{4}{\kappa +1}\right)}^{m-1}\hfill \\ & =& \frac{2\left(\kappa +1\right)}{\kappa \left(\kappa -3\right)},\left(\kappa >3\right).\hfill \end{array}$$

(7)

Furthermore, using the inequality $m{\left(\kappa \right)}^m\le 2^{m-1}{\left(\kappa \right)}_m,$ then we have

$$\begin{array}{ccc}\hfill \left|E_{\alpha ,\kappa }^{\prime }\left(z\right)\right|& =& \left|1+\underset{m=1}{\sum ^{\infty }}\left(m+1\right)\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}{z}^m\right|\hfill \\ & \ge & 1-\underset{m=1}{\sum ^{\infty }\left(m+1\right)}\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}\hfill \\ & \ge & 1-\underset{m=1}{\sum ^{\infty }}\frac{1}{{\left(\kappa \right)}_m}\hfill \\ & \ge & 1-\frac{2}{\kappa }\underset{m=1}{\sum ^{\infty }}{\left(\frac{2}{\kappa +1}\right)}^{m-1}\hfill \\ & =& \frac{{\kappa }^2-3\kappa -2}{\kappa \left(\kappa -1\right)}\left(\kappa >0\right).\hfill \end{array}$$

(8)

From (7) and (8), we get

$$\left|\frac{z{E}_{\alpha ,\kappa }^{″}\left(z\right)}{E_{\alpha ,\kappa }^{\prime }\left(z\right)}\right|\le \frac{2\left({\kappa }^2-1\right)}{\left(\kappa -1\right)\left({\kappa }^2-3\kappa -2\right)}.$$

(9)

This implies that $E_{\alpha ,\kappa }\left(z\right)\in \mathcal{C}$ in $\mathcal{D}$ if $\frac{2\left({\kappa }^2-1\right)}{\left(\kappa -1\right)\left({\kappa }^2-3\kappa -2\right)}\le 1$. To prove our result, we have to show that ${\kappa }^3-6{\kappa }^2+7\kappa +6\ge 0$. The inequality is satisfied for $\kappa \ge 3.5615528$. Hence, $E_{\alpha ,\kappa }\left(z\right)$ is convex in $\mathcal{D}$. □

Consider the integral operator ${\mathcal{F}}_{\gamma }:\mathcal{D}\to \mathbb{C}$, where $\gamma \in \mathbb{C},$ $\gamma \ne 0,$

$${\mathcal{F}}_{\gamma }\left(z\right)=\left\{\gamma \underset{0}{\stackrel{z}{\int }}{t}^{\gamma -2}{E}_{\alpha ,\kappa }\left(t\right)dt\right\},z\in \mathcal{D}.$$

Here, ${\mathcal{F}}_{\gamma }\in \mathcal{A}.$ We prove that ${\mathcal{F}}_{\gamma }\in \mathcal{S}$ in $\mathcal{D}.$

Theorem 8.

Let$M\in {\mathbb{R}}^{+}$such that$\left|E_{\alpha ,\kappa }\left(z\right)\right|\le M$in$\mathcal{D}$. If

$$\left|\gamma -1\right|+\frac{\kappa \left(\kappa +1\right)}{\left(\kappa -1\right)\left({\kappa }^2-\kappa -1\right)}+\frac{M}{\left|\gamma \right|}\le 1,$$

then${\mathcal{F}}_{\gamma }\in \mathcal{S}$in$\mathcal{D}.$

Proof. 

A calculation gives

$$\frac{z{\mathcal{F}}_{\gamma }^{″}\left(z\right)}{{\mathcal{F}}_{\gamma }^{\prime }\left(z\right)}=\frac{z{E}_{\alpha ,\kappa }^{\prime }\left(z\right)}{E_{\alpha ,\kappa }\left(z\right)}+\frac{z^{\gamma -1}}{\gamma }{E}_{\alpha ,\kappa }\left(z\right)+\gamma -2,z\in \mathcal{D}.$$

Since $E_{\alpha ,\kappa }\left(z\right)\in \mathcal{A}$, then by Schwarz Lemma, triangle inequality and (6), we obtain

$$\begin{array}{ccc}\hfill \left(1-{\left|z\right|}^2\right)\frac{z{\mathcal{F}}_{\gamma }^{″}\left(z\right)}{{\mathcal{F}}_{\gamma }^{\prime }\left(z\right)}& \le & \left(1-{\left|z\right|}^2\right)\left[\left|\gamma -1\right|+\left|\frac{z{E}_{\alpha ,\kappa }^{\prime }\left(z\right)}{E_{\alpha ,\kappa }\left(z\right)}-1\right|+\frac{{\left|z\right|}^{\gamma -1}}{\left|\gamma \right|}\left|\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\right|\right]\hfill \\ & \le & \left(1-{\left|z\right|}^2\right)\left[\left|\gamma -1\right|+\frac{\kappa \left(\kappa +1\right)}{\left(\kappa -1\right)\left({\kappa }^2-\kappa -1\right)}+\frac{M}{\left|\gamma \right|}\right].\hfill \end{array}$$

By using Lemma 2, ${\mathcal{F}}_{\gamma }\in \mathcal{S}$ in $\mathcal{D}$. □

Theorem 9.

Let$\alpha \ge 1,$$\mu \in \left[0,1\right)$and$z\in \mathcal{D}.$

(i)

If$\kappa >\frac{\left(11-3\mu \right)+\sqrt{{\mu }^2-12\mu +17}}{2\left(1-\mu \right)},$then$E_{\alpha ,\kappa }\left(z\right)\in \mathcal{K}\left(\frac{1+\mu }{2}\right).$

(ii)

If$\mu <1-\frac{\left\{(\kappa +2)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)+\left(\kappa +1\right)\right\}}{\kappa \left(\kappa +1\right)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)},$then$\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\in \mathcal{P}\left(\mu \right).$

(iii)

If$\left(1-\mu \right){\kappa }^3+\left(2\mu -3\right){\kappa }^2-\kappa +\left(1-\mu \right)>0,$then$E_{\alpha ,\kappa }\left(z\right)\in {\mathcal{S}}^{\ast }\left(\mu \right).$

(iv)

If$\left(1-\mu \right){\kappa }^3+\left(6\mu -8\right){\kappa }^2+\left(7-7\mu \right)\kappa +\left(8-6\mu \right)>0,$then$E_{\alpha ,\kappa }\left(z\right)\in \mathcal{C}\left(\mu \right).$

Proof. 

(i) Using (7) and Lemma 1, we get

$$\left|z{E}_{\alpha ,\kappa }^{″}\left(z\right)\right|\le \frac{2\left(\kappa +1\right)}{\kappa \left(\kappa -3\right)}<\frac{1-\mu }{4},$$

where $0\le \mu <1-\frac{8\left(\kappa +1\right)}{\kappa \left(\kappa -3\right)}$ and $\kappa >\frac{\left(11-3\mu \right)+\sqrt{{\mu }^2-12\mu +17}}{2\left(1-\mu \right)}.$ This shows that $E_{\alpha ,\kappa }\left(z\right)\in \mathcal{K}\left(\frac{1+\mu }{2}\right).$

(ii) To prove $\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\in \mathcal{P}\left(\mu \right),$ we have to show that $\left|g\left(z\right)-1\right|<1,$ where $g\left(z\right)=\frac{\{E_{\alpha ,\kappa }\left(z\right)/z\}-\mu }{\left(1-\mu \right)}.$ By using triangle inequality with

$$\begin{array}{ccc}\hfill \frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}& \le & \frac{1}{{\left(\kappa \right)}_m},m\in \mathbb{N},\hfill \\ \hfill {\left(\kappa \right)}_m& >& \kappa {\left({\kappa +\alpha }_0\right)}^{m-1},\left(\kappa >0;m\in \mathbb{N}\setminus \{1,2\}\right),\hfill \end{array}$$

(see [16]), where

$${\alpha }_0\approx 1.302775637\dots$$

is the largest root of the equation

$${\alpha }^2+\alpha -3=0,$$

we have

$$\begin{array}{ccc}\hfill \left|g\left(z\right)-1\right|& =& \left|\frac{1}{\left(1-\mu \right)}\sum _{m=1}^{\infty }\frac{\Gamma \left(\kappa \right)}{\Gamma \left(\alpha m+\kappa \right)}{z}^m\right|\hfill \\ & \le & \frac{1}{\left(1-\mu \right)}\sum _{m=1}^{\infty }\frac{1}{{\left(\kappa \right)}_m}\hfill \\ & \le & \frac{1}{\left(1-\mu \right)}\left\{\frac{1}{\kappa }+\frac{1}{\kappa \left(\kappa +1\right)}+\sum _{m=3}^{\infty }\frac{1}{\kappa {\left({\kappa +\alpha }_0\right)}^{m-1}}\right\}\hfill \\ & =& \frac{1}{\left(1-\mu \right)}\frac{\left\{(\kappa +2)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)+\left(\kappa +1\right)\right\}}{\kappa \left(\kappa +1\right)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)}.\hfill \end{array}$$

This implies that $\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\in \mathcal{P}\left(\mu \right),$ for $0<\mu <1-\frac{\left\{(\kappa +2)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)+\left(\kappa +1\right)\right\}}{\kappa \left(\kappa +1\right)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)}.$

(iii) We use the inequality $\left|\frac{z{E}_{\alpha ,\kappa }^{\prime }\left(z\right)}{E_{\alpha ,\kappa }\left(z\right)}-1\right|<1-\mu$ to show the starlikeness of order $\mu$ for the function $E_{\alpha ,\kappa }\left(z\right).$ By using (4) and (5), we have

$$\left|\frac{z{E}_{\alpha ,\kappa }^{\prime }\left(z\right)}{E_{\alpha ,\kappa }\left(z\right)}-1\right|\le \frac{\kappa \left(\kappa +1\right)}{\left(\kappa -1\right)\left({\kappa }^2-\kappa -1\right)}<1-\mu .$$

This implies that

$$\mu <1-\frac{\kappa \left(\kappa +1\right)}{\left(\kappa -1\right)\left({\kappa }^2-\kappa -1\right)}.$$

This completes the proof.

(iv) We use the inequality $\left|\frac{z{E}_{\alpha ,\kappa }^{″}\left(z\right)}{E_{\alpha ,\kappa }^{\prime }\left(z\right)}\right|<1-\mu$ to show that $E_{\alpha ,\kappa }\left(z\right)\in \mathcal{C}\left(\mu \right).$ By using (7) and (8), we have

$$\left|\frac{z{E}_{\alpha ,\kappa }^{″}\left(z\right)}{E_{\alpha ,\kappa }^{\prime }\left(z\right)}\right|\le \frac{2\left({\kappa }^2-1\right)}{\left(\kappa -3\right)\left({\kappa }^2-3\kappa -2\right)}<1-\mu .$$

This implies that

$$\mu <1-\frac{2\left({\kappa }^2-1\right)}{\left(\kappa -3\right)\left({\kappa }^2-3\kappa -2\right)},$$

hence the result. □

Substituting $\mu =0$ in Theorem 9, we obtained the following results.

Corollary 1.

Let$\alpha \ge 1,$$z\in \mathcal{D}$.

(i)

If$\kappa >\frac{11+\sqrt{17}}{2},$then$E_{\alpha ,\kappa }\left(z\right)\in \mathcal{K}\left(\frac{1}{2}\right).$

(ii)

If$\frac{\left\{(\kappa +2)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)+\left(\kappa +1\right)\right\}}{\kappa \left(\kappa +1\right)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)}<1,$then$\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\in \mathcal{P}.$

(iii)

If${\kappa }^3-3{\kappa }^2-\kappa +1>0,$then$E_{\alpha ,\kappa }\left(z\right)\in {\mathcal{S}}^{\ast }.$

(iv)

If${\kappa }^3-8{\kappa }^2+7\kappa +8>0,$then$E_{\alpha ,\kappa }\left(z\right)\in \mathcal{C}.$

Remark 2.

It is clear that$E_{\alpha ,\kappa }\left(z\right)\in \mathcal{K}\left(\frac{1}{2}\right)$when$\alpha \ge 1,$$\kappa >7.56155$and$E_{\alpha ,\kappa }\left(z\right)\in \mathcal{C}$when$\alpha \ge 1,$$\kappa >6.796963.$It concludes that our results improve the results of ([17], corollary 2.1).

3. Hardy Space of Mittag–Leffler Function

Consider the class $\mathcal{H}$ of analytic functions in $\mathcal{D}=\left\{z:\left|z\right|<1\right\}$ and ${\mathcal{H}}^{\infty }$ denote the space bounded functions on $\mathcal{H}$. Let $g\in \mathcal{H}$, set

$$M_q\left(r,g\right)=\left\{\begin{array}{c}{\left(\frac{1}{2\pi }\stackrel{2\pi }{\underset{0}{\int }}{\left|g\left(r{e}^{i\theta }\right)\right|}^{q}d\theta \right)}^{1/q},0<q<\infty ,\\ max\left\{\left|g\left(z\right)\right|:\left|z\right|\le r\right\},q=\infty .\end{array}\right.$$

If $M_q\left(r,g\right)$ is bounded for $r\in \left[0,1\right),$ then $g\in$ ${\mathcal{H}}^q$. It is clear that

$${\mathcal{H}}^{\infty }\subset {\mathcal{H}}^p\subset {\mathcal{H}}^q,0<p<q<\infty .$$

For some details, see [18]. It is also known [18] that, if $Re\left(g^{\prime }\left(z\right)\right)>0$ in $\mathcal{D}$, then

$$\left\{\begin{array}{c}{g}^{\prime }\in {\mathcal{H}}^p,p<1,\\ g\in {\mathcal{H}}^{p/\left(1-p\right)},0<p<1.\end{array}\right.$$

Hardy spaces of certain special functions are studied in [10,19,20].

Lemma 6

([21]). ${\mathcal{P}}_0\left(\mu \right)\ast {\mathcal{P}}_0\left(\eta \right)\subset {\mathcal{P}}_0\left(\gamma \right),$ where $\gamma =1-2\left(1-\mu \right)\left(1-\eta \right)$ and $\mu ,\eta <1.$ The value γ can not be improved.

Lemma 7

([22]). For $\mu ,\eta <1$ and $\gamma =1-2\left(1-\mu \right)\left(1-\eta \right),$ we have ${\mathcal{R}}_0\left(\mu \right)\ast {\mathcal{R}}_0\left(\eta \right)\subset {\mathcal{R}}_0\left(\gamma \right),$ or equivalently ${\mathcal{P}}_0\left(\mu \right)\ast {\mathcal{P}}_0\left(\eta \right)\subset {\mathcal{P}}_0\left(\gamma \right).$

Lemma 8

([23]). If the function g, convex of order μ, where $\mu \in \left[0,1\right)$, is not of the form

$$g\left(z\right)=\left\{\begin{array}{c}l+dz{\left(1-z{e}^{i\varsigma }\right)}^{2\mu -1},\mu \ne 1/2,\\ l+dlog\left(1-z{e}^{i\varsigma }\right),\mu =1/2,\end{array}\right.$$

for $d,l\in \mathbb{C}$, and $\varsigma \in \mathbb{R},$ then the following statements are true:

(i)

There exist$\delta =\delta \left(g\right)>0$such that$g^{\prime }\in {\mathcal{H}}^{\delta +1/\left[2\left(1-\mu \right)\right]}.$

(ii)

If$\mu \in \left[0,1/2\right),$then there exists$\tau =\tau \left(g\right)>0$such that$g\in {\mathcal{H}}^{\tau +1/\left(1-2\mu \right)}.$

(iii)

If$\mu \ge 1/2,$then$g\in {\mathcal{H}}^{\infty }.$

Theorem 10.

Let$\mu \in \left[0,1\right)$, $\left(1-\mu \right){\kappa }^3+\left(6\mu -8\right){\kappa }^2+\left(7-7\mu \right)\kappa +\left(8-6\mu \right)>0.$

(i)

If$\mu \in \left[0,1/2\right),$then$E_{\alpha ,\kappa }\left(z\right)\in {\mathcal{H}}^{1/\left(1-2\mu \right)}.$

(ii)

If$\mu \ge 1/2,$then$E_{\alpha ,\kappa }\left(z\right)\in {\mathcal{H}}^{\infty }.$

Proof. 

By using the definition of the hypergeometric function

$${}_2{F}_1\left(a,b,c;z\right)=\underset{m=0}{\sum ^{\infty }}\frac{{\left(a\right)}^m{\left(b\right)}^m}{{\left(c\right)}^m}\frac{z^m}{m!},$$

we have

$$\begin{array}{ccc}\hfill l+\frac{dz}{{\left(1-z{e}^{i\varsigma }\right)}^{1-2\mu }}& =& l+d{z}_2{F}_1\left(1,1-2\alpha ,1;z{e}^{i\varsigma }\right),\hfill \\ & =& l+d\underset{m=0}{\sum ^{\infty }}\frac{{\left(1-2\alpha \right)}_m}{m!}{e}^{i\varsigma m}{z}^{m+1},\hfill \end{array}$$

for $l,d\in \mathbb{C},$ $\mu \ne 1/2$ and for real $\varsigma .$ On the other hand,

$$\begin{array}{ccc}\hfill l+dlog\left(1-z{e}^{i\gamma }\right)& =& l-d{z}_2{F}_1\left(1,1,2;z{e}^{i\varsigma }\right),\hfill \\ & =& l-d\underset{m=0}{\sum ^{\infty }}\frac{1}{m+1}{e}^{i\varsigma m}{z}^{m+1}.\hfill \end{array}$$

Therefore, the function $E_{\alpha ,\kappa }\left(z\right)$ is not of the form of $l+dz{\left(1-z{e}^{i\gamma }\right)}^{2\mu -1}\left(\mathrm{for}\mu \ne 1/2\right)$ and $l+dlog\left(1-z{e}^{i\gamma }\right)$ $\left(\mathrm{for}\mu =1/2\right).$ We know that, by part (iv) of Theorem 9, $E_{\alpha ,\kappa }\left(z\right)\in \mathcal{C}\left(\mu \right).$ Therefore, by using Lemma 8, we have the required result. □

Theorem 11.

Let$\frac{\left\{(\kappa +2)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)+\left(\kappa +1\right)\right\}}{\kappa \left(\kappa +1\right)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)}<1$and$f\in \mathcal{D}.$Then, convolution$E_{\alpha ,\kappa }\ast f$is in${\mathcal{H}}^{\infty }\cap \mathcal{R}.$

Proof. 

Let $h\left(z\right)=E_{\alpha ,\kappa }\left(z\right)\ast g\left(z\right).$ Then, $h^{\prime }\left(z\right)=\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\ast g^{\prime }\left(z\right).$ Using the Corollary 1 part ii, we have $\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\in \mathcal{P}.$ As $g\in \mathcal{R}$; therefore, by using Lemma 6 $h\in \mathcal{R}.$ Now, the function $\frac{E_{\alpha ,\kappa }\left(z\right)}{z}$ is complete; therefore, $h\left(z\right)$ is complete. This implies that $h\left(z\right)$ is bounded. Thus, we have the required result. □

Theorem 12.

Let$\mu <1-\frac{\left\{(\kappa +2)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)+\left(\kappa +1\right)\right\}}{\kappa \left(\kappa +1\right)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)},$$\mu \in \left[0,1\right)$and$z\in \mathcal{D}.$If$g\in \mathcal{P}\left(\eta \right),$then$E_{\alpha ,\kappa }\left(z\right)\ast g\in \mathcal{R}\left(\gamma \right),$where$\gamma =1-2\left(1-\mu \right)\left(1-\eta \right).$

Proof. 

Let $h\left(z\right)=E_{\alpha ,\kappa }\left(z\right)\ast g\left(z\right).$ Then, it is clear that $h^{\prime }\left(z\right)=\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\ast g^{\prime }\left(z\right).$ Using Theorem 9 part (ii), we have $\frac{E_{\alpha ,\kappa }\left(z\right)}{z}\in \mathcal{P}\left(\mu \right).$ As $g\in \mathcal{R}$, therefore, by using Lemma 6 and the fact that $g^{\prime }\in \mathcal{P}\left(\eta \right),$ we have $h^{\prime }\left(z\right)\in \mathcal{P}\left(\gamma \right),$ where $\gamma =1-2\left(1-\mu \right)\left(1-\eta \right).$ Consequently, $h\in \mathcal{R}\left(\gamma \right).$ □

Corollary 2.

Let$\mu \in \left[0,1\right)$and$\mu <1-\frac{\left\{(\kappa +2)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)+\left(\kappa +1\right)\right\}}{\kappa \left(\kappa +1\right)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)}.$If$g\in \mathcal{R}\left(\eta \right)$,$\eta =\left(1-2\mu \right)/\left(2-2\mu \right),$then$E_{\alpha ,\kappa }\left(z\right)\ast g\in \mathcal{R}\left(0\right).$

Corollary 3.

Let$\mu \in \left[0,1\right)$and$\frac{\left\{(\kappa +2)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)+\left(\kappa +1\right)\right\}}{\kappa \left(\kappa +1\right)\left(\kappa +{\alpha }_0\right)\left(\kappa +{\alpha }_0-1\right)}<1.$If$g\in \mathcal{R}\left(1/2\right)$, then$E_{\alpha ,\kappa }\left(z\right)\ast g\in \mathcal{R}\left(0\right).$

4. Applications

Now, we present some applications of the above theorems. It is clear that

$$E_{1,2}\left(z\right)=e^z-1,E_{1,3}\left(z\right)=\frac{2e^z-z-1}{z},E_{1,4}\left(z\right)=\frac{6e^z-3z^2,-6z-6}{z^2}.$$

From Theorem 9, we get the following:

Corollary 4.

$\left(i\right)$If$0\le \mu <{\mu }_0,$where${\mu }_0\approx 0.26759,$then$E_{1,2}\left(z\right)\in \mathcal{P}\left(\mu \right).$

$\left(ii\right)$If$0\le \mu <{\mu }_1,$where${\mu }_1\approx 0.55988,$then$E_{1,3}\left(z\right)\in \mathcal{P}\left(\mu \right).$

$\left(iii\right)$If$0\le \mu <{\mu }_2,$where${\mu }_2\approx 0.68904,$then$E_{1,4}\left(z\right)\in \mathcal{P}\left(\mu \right).$

Corollary 5.

If$0\le \mu <{\mu }_3,$where${\mu }_3\approx 0.39393,$then$E_{1,4}\left(z\right)\in {\mathcal{S}}^{\ast }\left(\mu \right).$

Corollary 6.

$\left(i\right)$Let$0\le \mu <{\mu }_4$, where${\mu }_4\approx 0.2675930.$If$g\in \mathcal{R}\left(\eta \right)$,$\eta =\left(1-2\mu \right)/\left(2-2\mu \right),$then$E_{1,2}\left(z\right)\ast g\in \mathcal{R}\left(0\right).$

$\left(ii\right)$Let$0\le \mu <{\mu }_5$, where${\mu }_5\approx 0.55987780.$If$g\in \mathcal{R}\left(\eta \right)$,$\eta =\left(1-2\mu \right)/\left(2-2\mu \right),$then$E_{1,3}\left(z\right)\ast g\in \mathcal{R}\left(0\right).$

$\left(iii\right)$Let$0\le \mu <{\mu }_6$, where${\mu }_6\approx 0.68904320.$If$g\in \mathcal{R}\left(\eta \right)$,$\eta =\left(1-2\mu \right)/\left(2-2\mu \right),$then$E_{1,4}\left(z\right)\ast g\in \mathcal{R}\left(0\right).$

5. Conclusions

In this paper, we have studied certain geometric properties of Mittag-Leffler functions such as starlikeness, convexity and close-to-convexity. We have also found the Hardy spaces of Mittag-Leffler functions. Further, we have improved some results in the literature.