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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

From (7) and (8), we get

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

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.

Author Contributions

Conceptualization, S.N. and M.R.; Formal analysis, S.N. and M.R.; Funding acquisition, J.L.L.; Investigation, S.N.; Methodology, S.N. and M.R.; Supervision, M.R.; Validation, M.A.; Visualization, M.A.;Writing—original draft, S.N.; Writing—review and editing, J.L.L.

Funding

The work presented here is supported by the Natural Science Foundation of Jiangsu Province under Grant No. BK20151304 and the National Natural Science Foundation of China under Grant No. 11571299.

Acknowledgments

The authors are grateful to the referees for their valuable comments and suggestions which improve the presentation of paper.

Conflicts of Interest

The authors declare no conflict of interest.