On Geometric Properties of k-Mittag–Leffler Function

Abstract

This paper investigates geometric properties of a normalized k-Mittag–Leffler function, where $k>0$ is a fixed real parameter, involving the k-Gamma function and the k-Pochhammer symbol. Since the coefficient structure of this function differs essentially from that of the classical Mittag–Leffler and Le Roy-type Mittag–Leffler functions, the standard monotonicity arguments available in the literature cannot be applied directly. To overcome this difficulty, we first derive a two-sided inequality for the k-digamma function and use it to establish suitable monotonicity properties of the associated coefficient sequences. These auxiliary results form the main analytical foundation of the paper. By combining them with classical criteria from geometric function theory, we obtain explicit sufficient conditions for the normalized k-Mittag–Leffler function to be starlike, parabolic starlike, convex, uniformly convex, and close-to-convex in the unit disk. Corresponding starlikeness and convexity results in the half-unit disk are also established. In addition, we study an associated odd normalized function and derive conditions ensuring its close-to-convexity with respect to ${\displaystyle \frac{1}{2}}\log \left({\displaystyle \frac{1+\tau }{1-\tau }}\right)$. Several corollaries and examples are presented to analyze the applicability of the results. For $k=1$, the obtained assertions reduce to the corresponding results for the normalized Prabhakar (three-parameter) Mittag–Leffler function, and for $k=q=1$ to those for the normalized two-parameter Mittag–Leffler function.

1. Introduction

Let $D=\{\tau \in \mathbb{C}:|\tau |<1\}$ represent the open unit disk in the complex plane $\mathbb{C}$. Also let $\mathcal{H}$ represent the class of analytic functions (AFs) in D. Every function $\chi \in \mathcal{H}$ admits a Taylor-series representation

$$\chi \left(\tau \right)={\displaystyle \sum _{l=0}^{\infty }}{a}_l{\tau }^l,\tau \in D.$$

We use $\mathcal{A}$ to represent the normalized subclass of $\mathcal{H}$ consisting of series of the form

$$\chi \left(\tau \right)=\tau +{\displaystyle \sum _{l=2}^{\infty }}{a}_l{\tau }^l,\tau \in D,$$

(1)

or, equivalently,

$$\chi \left(0\right)=0\mathrm{and}{\chi }^{\prime }\left(0\right)=1.$$

The class of all functions in $\mathcal{A}$ that are one-to-one in D is denoted by $\mathcal{S}$. Such functions are called univalent functions. The systematic study of univalent functions and their coefficient, growth, distortion, and covering properties forms a central part of geometric function theory; standard accounts can be found in [1,2].

A function $\chi \in \mathcal{A}$ is starlike with respect to the origin if $\chi \left(D\right)$ is a domain that is starlike with respect to the origin. The class of such functions is denoted by ${\mathcal{S}}^{*}$. Its standard analytic characterization is

$$\chi \in {\mathcal{S}}^{*}⟺\chi \in \mathcal{A}:\mathfrak{R}\left({\displaystyle \frac{\tau {\chi }^{\prime }\left(\tau \right)}{\chi \left(\tau \right)}}\right)>0,\tau \in D.$$

(2)

More generally, for $0\le \delta <1$, a function $\chi \in \mathcal{A}$ is referred to as starlike of order $\delta$ if

$$\mathfrak{R}\left({\displaystyle \frac{\tau {\chi }^{\prime }\left(\tau \right)}{\chi \left(\tau \right)}}\right)>\delta ,\tau \in D.$$

(3)

The corresponding class is denoted by ${\mathcal{S}}^{*}\left(\delta \right)$ (see [1,2]).

A function $\chi \in \mathcal{A}$ is said to be convex in D if the image $\chi \left(D\right)$ is a convex domain. The class of such functions is denoted by $\mathcal{K}$ and is characterized analytically by

$$\chi \in \mathcal{K}⟺\chi \in \mathcal{A}:\mathfrak{R}\left(1+{\displaystyle \frac{\tau {\chi }^{″}\left(\tau \right)}{{\chi }^{\prime }\left(\tau \right)}}\right)>0,\tau \in D.$$

(4)

Equivalently,

$$\chi \in \mathcal{K}⟺\tau {\chi }^{\prime }\left(\tau \right)\in {\mathcal{S}}^{*},$$

(5)

which is the classical Alexander relation.

Fix $0\le \delta <1$. The class $\mathcal{K}\left(\delta \right)$ denotes the collection of convex functions of order $\delta$, namely those $\chi \in \mathcal{A}$ satisfying

$$\mathfrak{R}\left(1+{\displaystyle \frac{\tau {\chi }^{″}\left(\tau \right)}{{\chi }^{\prime }\left(\tau \right)}}\right)>\delta ,\tau \in D.$$

(6)

A function $\chi \in \mathcal{A}$ is called close-to-convex (see [1,3]) in D if there exist a $g\in {\mathcal{S}}^{*}$ and $\beta \in \mathbb{R}$ such that

$$\mathfrak{R}\left(e^{i\beta }{\displaystyle \frac{\tau {\chi }^{\prime }\left(\tau \right)}{g\left(\tau \right)}}\right)>0,\tau \in D.$$

(7)

Goodman [4,5] established the class of uniformly convex functions and the class of uniformly starlike functions, which have subsequently been important in the geometric theory of normalized special functions.

A function $\chi \in \mathcal{A}$ is uniformly convex if the image under $\chi$ of every circular arc contained in D, whose center also belongs to D, is a convex arc. The corresponding class is denoted by $\mathcal{UCV}$. Rønning [6] investigated a related class of parabolic starlike functions, denoted by ${\mathcal{S}}_p$, which may be expressed as

$${\mathcal{S}}_p=\left\{\chi \in \mathcal{A}:\chi \left(\tau \right)=\tau F^{\prime }\left(\tau \right),F\in \mathcal{UCV}\right\}.$$

If $f\in {\mathcal{S}}_p$, then for all $z\in D$, the value $z{f}^{\prime }\left(z\right)/f\left(z\right)$ lies in a domain in the right half-plane bounded by a parabola with focus at $z=1$.

The Gamma function is a cornerstone of special function theory and generalizes the factorial function to the complex domain. For $\mathfrak{R}\left(\upsilon \right)>0$, it is represented by

$$\begin{array}{cc}\hfill \Gamma \left(\upsilon \right)& ={\int }_0^{\infty }{e}^{-\xi }{\xi }^{\upsilon -1}d\xi \hfill \\ \hfill & ={\displaystyle \frac{1}{\upsilon }}{\displaystyle \prod _{n=1}^{\infty }}{\displaystyle \frac{{\left(1+\frac{1}{n}\right)}^{\upsilon }}{1+\frac{\upsilon }{n}}}\hfill \\ \hfill & =\underset{n\to \infty }{\lim }{\displaystyle \frac{n!n^{\upsilon }}{\upsilon (\upsilon +1)(\upsilon +2)\cdots (\upsilon +n)}}.\hfill \end{array}$$

(8)

and satisfies the recurrence relation

$$\Gamma (\upsilon +1)=\upsilon \Gamma \left(\upsilon \right).$$

(9)

The Gamma function extends meromorphically to the complex plane, with simple poles at $0,-1,-2,\dots$. It appears naturally in the coefficients of many important special functions, including hypergeometric, Bessel, Wright, and Mittag–Leffler functions [7,8]. In 1903, Mittag–Leffler [9,10] introduced the one-parameter function

$$E_{\sigma }\left(\tau \right)={\displaystyle \sum _{l=0}^{\infty }}{\displaystyle \frac{{\tau }^l}{\Gamma (\sigma l+1)}},\tau \in \mathbb{C},$$

(10)

where $\sigma >0$ is a parameter. This function generalizes the exponential function, since $E_1\left(\tau \right)=e^{\tau }$. The generalization of the Mittag–Leffler (ML) function to two parameters $\sigma ,\nu >0$ is defined as

$$E_{\sigma ,\nu }\left(\tau \right)={\displaystyle \sum _{l=0}^{\infty }}{\displaystyle \frac{{\tau }^l}{\Gamma (\sigma l+\nu )}},\tau \in \mathbb{C}.$$

(11)

Al-Bassam and Luchko [11] formulated a multi-index Mittag–Leffler function connected with products of Gamma functions and applied it to fractional integro-differential equations. Bansal and Prajapat [12] and Mehrez and Das [13] investigated several geometric characteristics of normalized Mittag–Leffler-type functions.

Le Roy [14] introduced the function

$$R_{\gamma }\left(\tau \right)={\displaystyle \sum _{l=0}^{\infty }}{\displaystyle \frac{{\tau }^l}{{(l!)}^{\gamma }}},\tau \in \mathbb{C},$$

(12)

where $\gamma >0,$ in connection with the asymptotic behavior of certain power series.

Special functions constitute an important branch of mathematical analysis and play a fundamental role in complex analysis, fractional calculus, approximation theory, mathematical physics, engineering, and the theory of differential and integral equations. The integral representations of ML functions investigated by Gerhold [15] and Garra and Polito [16] established Le Roy-type generalizations of ML functions. Garrappa et al. [17] studied the Laplace transforms, and asymptotic expansions, while Górska et al. [18] established complete monotonicity properties. In a parallel direction, Díaz and Pariguan [19] introduced the k-Pochhammer symbol and the associated k-Gamma function, laying the foundation for k-analogue theory. Building on this framework, Dorrego and Cerutti [20] introduced the generalized k-ML function, thereby extending the classical ML family. More recently, the geometric properties of normalized special functions have been extensively studied. In 2022, Das and Mehrez [21] investigated the normalized four-parameter Wright function and established geometric properties such as starlikeness, convexity, uniformly convexity, and close-to-convexity, together with applications to normalized Bessel and two-parameter Wright functions. In 2024, Gangania and Kazımoğlu [22] studied geometric properties of functions containing derivatives of Bessel functions. In 2023, Naz et al. [23] studied generalized Bessel functions linked to the exponential function using differential-subordination techniques, while Tomovski et al. [24] established geometric features of generalized Mathieu-type power series by applying classical coefficient criteria. Subsequently, Alarifi and Mondal [25] studied geometric characteristics of Bessel–Struve kernel functions. Recent advances in geometric function theory have further expanded the study of normalized special functions and their applications. Khan et al. [26] investigated several applications of AFs related to the q-fractional operator and established various geometric features within the framework of quantum calculus. Srivastava et al. [27] derived Faber polynomial coefficient inequalities for bi-Bazilevič functions related to the Fibonacci number and square-root series. More recently, Abubakr et al. [28] studied the geometric features and Hardy spaces of the normalized Miller–Ross function, while Ali and Raza [29] obtained coefficient difference estimates for subclasses of non-Bazilevič functions. See for more details [30,31,32]. These contributions further demonstrate the growing interest in investigating the geometric behavior of special functions and related analytic function classes using operator-theoretic, coefficient-based, and functional-analytic techniques. Despite these significant developments, the geometric properties of normalized k-Mittag–Leffler functions have received comparatively little attention. In particular, the coefficient techniques developed for classical ML and Le Roy-type ML functions cannot be applied directly because the coefficients simultaneously involve the k-Gamma function and the k-Pochhammer symbol. Consequently, new analytical tools are required to establish the monotonicity of the corresponding coefficient sequences.

Throughout this paper, $k>0$ is a fixed real parameter and $\log k$ denotes its real logarithm. The k-Pochhammer symbol ${\left(\upsilon \right)}_{r,k}$ was introduced by Díaz and Pariguan [19] and is defined by

$${\left(\upsilon \right)}_{0,k}=1,{\left(\upsilon \right)}_{r,k}=\upsilon (\upsilon +k)\cdots (\upsilon +(r-1)k),r\in \mathbb{N},\upsilon \in \mathbb{C}.$$

(13)

If $\upsilon \notin -k{\mathbb{N}}_0=\{0,-k,-2k,\dots \}$, then (see Remark 1 below)

$${\left(\upsilon \right)}_{r,k}={\displaystyle \frac{{\Gamma }_k(\upsilon +rk)}{{\Gamma }_k\left(\upsilon \right)}},r\in {\mathbb{N}}_0,$$

(14)

and the right-hand side of (14) is used to define ${\left(\upsilon \right)}_{s,k}$ for complex s with $\upsilon +sk\notin -k{\mathbb{N}}_0$. In particular, for $q>0$ the function $s\mapsto {\left(q\right)}_{s,k}={\Gamma }_k(q+sk)/{\Gamma }_k\left(q\right)$ is a smooth positive function on $[0,\infty )$ which coincides with (13) at the integers. Note that for $k=1$ these reduce to the classical Pochhammer symbol

$${\left(\upsilon \right)}_{r,1}={\left(\upsilon \right)}_r=\upsilon (\upsilon +1)\cdots (\upsilon +(r-1)),r\in \mathbb{N},{\left(\upsilon \right)}_0=1.$$

(15)

The k-Gamma function ${\Gamma }_k$ is related to the classical Gamma function $\Gamma$ by

$${\Gamma }_k\left(\upsilon \right)=k^{{\displaystyle \frac{\upsilon }{k}}-1}\Gamma \left({\displaystyle \frac{\upsilon }{k}}\right),\mathfrak{R}\left(\upsilon \right)>0,k>0.$$

(16)

Definition 1. 

For $k>0$ the k-digamma function ${\psi }_k\left(\upsilon \right)$ is given by

$${\psi }_k\left(\upsilon \right)={\displaystyle \frac{{\Gamma }_k^{\prime }\left(\upsilon \right)}{{\Gamma }_k\left(\upsilon \right)}},\upsilon >0.$$

(17)

For $k=1$, the k-digamma function ${\psi }_k$ reduces to the classical digamma function

$$\psi \left(\upsilon \right)={\Gamma }^{\prime }\left(\upsilon \right)/\Gamma \left(\upsilon \right),\upsilon >0.$$

(18)

Proposition 1 

([19]). If $k>0$ then the k-Gamma function satisfies the recurrence relation

$${\Gamma }_k(\upsilon +k)=\upsilon {\Gamma }_k\left(\upsilon \right),\upsilon \notin -k{\mathbb{N}}_0.$$

(19)

Remark 1. 

Since $k>0$ (see the standing convention at the beginning of Section 1), the quantity $\log k$ denotes the real logarithm of the positive number k, and hence

$$k^{{\displaystyle \frac{\upsilon }{k}}-1}=\exp \left(\left({\displaystyle \frac{\upsilon }{k}}-1\right)\log k\right)$$

is single-valued. Moreover, the classical Gamma function Γ is understood in its standard analytic continuation on

$$\mathbb{C}\setminus \{0,-1,-2,\dots \},$$

so that $\Gamma (\upsilon /k)$ is well defined whenever $\mathfrak{R}\left(\upsilon \right)>0$. Since Γ has no zeros, the right-hand side of (16) defines the meromorphic continuation of ${\Gamma }_k$ to $\mathbb{C}$; its poles are exactly the points $\upsilon \in -k{\mathbb{N}}_0=\{0,-k,-2k,\dots \}$, ${\Gamma }_k$ has no zeros, and therefore $1/{\Gamma }_k$ is entire. The recurrence (19) follows from (16) and $\Gamma (z+1)=z\Gamma \left(z\right)$ for all $\upsilon \notin -k{\mathbb{N}}_0$. For real $\upsilon >0$ we have ${\Gamma }_k\left(\upsilon \right)>0$. In all main results of this paper only positive arguments of ${\Gamma }_k$ and ${\psi }_k$ occur.

By the definition (13) of the k-Pochhammer symbol, for $r=l\in \mathbb{N}$ and $\upsilon =q$, we have

$${\left(q\right)}_{l,k}=q(q+k)(q+2k)\cdots (q+(l-1)k).$$

(20)

Similarly, taking $r=l-1,l\ge 2,l\in \mathbb{N}$ in (13), we obtain

$${\left(q\right)}_{l-1,k}=q(q+k)(q+2k)\cdots (q+(l-2)k).$$

(21)

On multiplying both sides of (21) by $q+(l-1)k$, we get

$$\begin{array}{cc}\hfill {\left(q\right)}_{l-1,k}\left(q+(l-1)k\right)& =q(q+k)(q+2k)\cdots (q+(l-2)k)\left(q+(l-1)k\right)\hfill \\ \hfill & =q(q+k)(q+2k)\cdots (q+(l-1\left)k\right)\hfill \\ \hfill & ={\left(q\right)}_{l,k},\hfill \end{array}$$

(22)

which proves the recurrence relation

$${\left(q\right)}_{l,k}={\left(q\right)}_{l-1,k}\left(q+(l-1)k\right),l\in \mathbb{N},l\ge 2.$$

(23)

The following integral representation holds for the k-Gamma function:

$${\Gamma }_k\left(\upsilon \right)={\int }_0^{\infty }{e}^{-{\xi }^k/k}{\xi }^{\upsilon -1}d\xi ,k>0,\mathfrak{R}\left(\upsilon \right)>0.$$

(24)

From (14) and (16), we obtain, for $\upsilon \notin -k{\mathbb{N}}_0$ and $r\in {\mathbb{N}}_0$,

$${\left(\upsilon \right)}_{r,k}={\displaystyle \frac{k^{\frac{\upsilon }{k}+r-1}\Gamma \left(\frac{\upsilon }{k}+r\right)}{k^{\frac{\upsilon }{k}-1}\Gamma \left(\frac{\upsilon }{k}\right)}}=k^r{\left({\displaystyle \frac{\upsilon }{k}}\right)}_r.$$

(25)

In ref. [20], Dorrego and Cerutti introduced the k-Mittag–Leffler function by

$$E_{k,\sigma ,\nu }^q\left(\tau \right)={\displaystyle \sum _{r=0}^{\infty }}{\displaystyle \frac{{\left(q\right)}_{r,k}{\tau }^r}{{\Gamma }_k(\sigma r+\nu )r!}},k>0,\min \{\mathfrak{R}\left(\sigma \right),\mathfrak{R}\left(\nu \right)\}>0,q,\sigma ,\tau \in \mathbb{C}.$$

(26)

Since ${\left(q\right)}_{0,k}=1$, we have $E_{k,\sigma ,\nu }^q\left(0\right)=1/{\Gamma }_k\left(\nu \right)$, so that $E_{k,\sigma ,\nu }^q\notin \mathcal{A}$. Because $\mathfrak{R}\left(\nu \right)>0$, Remark 1 gives $0<|{\Gamma }_k\left(\nu \right)|<\infty$, and hence the normalization below is well defined. We define the normalized k-Mittag–Leffler function by

$${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)=\tau {\Gamma }_k\left(\nu \right)E_{k,\sigma ,\nu }^q\left(\tau \right),k>0,\min \{\mathfrak{R}\left(\sigma \right),\mathfrak{R}\left(\nu \right)\}>0,q,\sigma ,\tau \in \mathbb{C}.$$

(27)

Substituting (26) into (27), we obtain

$${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)=\tau {\Gamma }_k\left(\nu \right){\displaystyle \sum _{r=0}^{\infty }}{\displaystyle \frac{{\left(q\right)}_{r,k}{\tau }^r}{{\Gamma }_k(\sigma r+\nu )r!}}={\displaystyle \sum _{r=0}^{\infty }}{\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{r,k}}{{\Gamma }_k(\sigma r+\nu )r!}}{\tau }^{r+1}.$$

Since

$${\left(q\right)}_{0,k}=1,$$

the first term of (28) becomes

$${\displaystyle \frac{{\Gamma }_k\left(\nu \right)}{{\Gamma }_k\left(\nu \right)}}\tau =\tau .$$

Thus

$${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)=\tau +{\displaystyle \sum _{r=1}^{\infty }}{\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{r,k}}{{\Gamma }_k(\sigma r+\nu )r!}}{\tau }^{r+1}.$$

(28)

Replacing $r+1=l$, we obtain

$${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)=\tau +{\displaystyle \sum _{l=2}^{\infty }}{a}_l(k,\sigma ,\nu ,q){\tau }^l,$$

(29)

where

$$a_l(k,\sigma ,\nu ,q)={\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{l-1,k}}{{\Gamma }_k(\sigma (l-1)+\nu )(l-1)!}},l\ge 2.$$

(30)

Formula (29) holds for

$$k>0,\min \left\{\mathfrak{R}\right(\sigma ),\mathfrak{R}(\nu \left)\right\}>0,q,\sigma \in \mathbb{C},\tau \in \mathbb{C};$$

throughout, however, we consider only positive real parameters

$$k,\sigma ,\nu ,q>0,$$

so that ${\Gamma }_k\left(\nu \right)\in (0,\infty )$ and $a_l(k,\sigma ,\nu ,q)>0$ for all $l\ge 2$.

Lemma 1. 

Let $k,\sigma ,\nu ,q>0$. Then the series in (29) converges for all $\tau \in \mathbb{C}$. In particular, ${\mathcal{E}}_{k,\sigma ,\nu }^q\in \mathcal{A}$.

Proof. 

By (13), (16) and (30),

$${\displaystyle \frac{a_{l+1}}{a_l}}={\displaystyle \frac{q+(l-1)k}{l}}·{\displaystyle \frac{{\Gamma }_k(\sigma (l-1)+\nu )}{{\Gamma }_k(\sigma l+\nu )}}={\displaystyle \frac{q+(l-1)k}{l}}·{\displaystyle \frac{k^{-\sigma /k}\Gamma \left(x_l\right)}{\Gamma (x_l+\sigma /k)}},x_l={\displaystyle \frac{\sigma (l-1)+\nu }{k}}.$$

Since $\Gamma (x+a)/\Gamma \left(x\right)\sim x^a$ as $x\to \infty$, we get $a_{l+1}/a_l\sim k{\left(\sigma l\right)}^{-\sigma /k}\to 0$ as $l\to \infty$. Hence the radius of convergence is infinite. □

Remark 2. 

For $k=1$, $E_{1,\sigma ,\nu }^q=E_{\sigma ,\nu }^q$ is the Prabhakar (three-parameter) Mittag–Leffler function, and for $k=q=1$, since ${\left(1\right)}_r=r!$, it reduces to the two-parameter function $E_{\sigma ,\nu }$ in (11).

1.1. Main Contributions, Research Gap and Novelty

Although several geometric properties of classical ML functions and their variants have been investigated in recent years, the corresponding theory for normalized k-Mittag–Leffler functions remains largely unexplored. In particular, the geometric function theoretic framework established by Mehrez and Das [13] for the normalized Le Roy-type ML function cannot be extended directly to the k-Mittag–Leffler setting because the coefficients involve both the k-Gamma function and the k-Pochhammer symbol. Consequently, the existing coefficient estimates, monotonicity arguments, and geometric criteria developed for the classical Gamma-function framework are no longer applicable. This constitutes the principal research gap addressed in the present work.

To bridge this gap, we first establish a new two-sided inequality for the k-digamma function and employ it to derive suitable monotonicity properties of the coefficient sequences associated with the normalized k-Mittag–Leffler function. These new auxiliary results constitute the main analytical tools of the paper and enable us to overcome the difficulties arising from the presence of the k-Gamma function.

Based on these results and several classical lemmas from geometric function theory, we obtain explicit sufficient conditions for the normalized k-Mittag–Leffler function to be starlike, convex, uniformly convex, parabolic starlike, and close-to-convex in the unit disk. We also establish corresponding starlikeness and convexity results in the half-unit disk and investigate the close-to-convexity of an associated odd normalized function with respect to

$${\displaystyle \frac{1}{2}}\log \left({\displaystyle \frac{1+\tau }{1-\tau }}\right).$$

Furthermore, several corollaries, special cases, and illustrative examples are presented to prove the applicability of the obtained criteria.

The novelty of the present paper lies in introducing a systematic geometric function theoretic study of the normalized k-Mittag–Leffler function by combining new k-digamma inequalities with coefficient monotonicity techniques. To the best of our knowledge, this is the first work that derives explicit geometric conditions for normalized k-Mittag–Leffler functions through the use of the k-digamma function. Moreover, our approach adapts the method of Mehrez and Das [13] from the classical Gamma-function framework to the k-Gamma setting. In the case $k=1$, the investigated results reduce to corresponding geometric properties of the normalized Prabhakar function and, for $k=q=1$, of the normalized two-parameter Mittag–Leffler function (cf. [12]).

1.2. Set of Lemmas

We derive the main results concerning the geometric characteristics of the normalized k-Mittag–Leffler function with the aid of the following lemmas.

Lemma 2 

([33]). Let $\chi \in \mathcal{A}$ and suppose that

$$\left|{\displaystyle \frac{\chi \left(\tau \right)}{\tau }}-1\right|<1,\tau \in D.$$

(31)

Then

$$\mathrm{Re}\left({\displaystyle \frac{\tau {\chi }^{\prime }\left(\tau \right)}{\chi \left(\tau \right)}}\right)>0,\left|\tau \right|<{\displaystyle \frac{1}{2}}.$$

(32)

This means that χ is univalent and starlike in

$$D_{1/2}=\left\{\tau :\left|\tau \right|<{\displaystyle \frac{1}{2}}\right\}.$$

Lemma 3 

([34]). Let $\chi \in \mathcal{A}$. If

$$|{\chi }^{\prime }\left(\tau \right)-1|<{\displaystyle \frac{2}{\sqrt{5}}},\tau \in D,$$

then χ is starlike in D.

Lemma 4 

([35]). Let $\chi \in \mathcal{A}$ and suppose that

$$|{\chi }^{\prime }\left(\tau \right)-1|<1,\tau \in D,$$

(33)

then χ is convex in

$$D_{1/2}=\left\{\tau :\left|\tau \right|<{\displaystyle \frac{1}{2}}\right\}.$$

Lemma 5 

([36]). Let $\chi \in \mathcal{A}$.

(i) 

If

$$\left|{\displaystyle \frac{\tau {\chi }^{″}\left(\tau \right)}{{\chi }^{\prime }\left(\tau \right)}}\right|<{\displaystyle \frac{1}{2}},\tau \in D,$$

(34)

then $\chi \in \mathcal{UCV}$.

(ii) 

If

$$\left|{\displaystyle \frac{\tau {\chi }^{\prime }\left(\tau \right)}{\chi \left(\tau \right)}}-1\right|<{\displaystyle \frac{1}{2}},\tau \in D,$$

(35)

then $\chi \in {\mathcal{S}}_p$.

Lemma 6 

([37]). Let

$$\chi \left(\tau \right)=\tau +{\displaystyle \sum _{l=2}^{\infty }}{A}_l{\tau }^l,\tau \in D.$$

If either

$$1\le 2A_2\le \cdots \le l{A}_l\le (l+1)A_{l+1}\le \cdots \le 2,$$

(36)

or

$$1\ge 2A_2\ge \cdots \ge l{A}_l\ge (l+1)A_{l+1}\ge \cdots \ge 0,$$

(37)

then χ is close-to-convex with respect to the starlike function $-\log (1-\tau ).$

Lemma 7 

([37]). Let

$$\chi \left(\tau \right)=\tau +{\displaystyle \sum _{n=2}^{\infty }}{A}_{2n-1}{\tau }^{2n-1}$$

be analytic in D. If either

$$1\ge 3A_3\ge \cdots \ge (2n-1)A_{2n-1}\ge \cdots \ge 0,$$

(38)

or

$$1\le 3A_3\le \cdots \le (2n-1)A_{2n-1}\le \cdots \le 2,$$

(39)

then χ is close-to-convex in D with respect to the convex function

$$h\left(\tau \right)={\displaystyle \frac{1}{2}}\log \left({\displaystyle \frac{1+\tau }{1-\tau }}\right),$$

that is, $\mathfrak{R}\left({\chi }^{\prime }\left(\tau \right)/h^{\prime }\left(\tau \right)\right)>0$ in D. In particular, χ is univalent in D.

Lemma 8 

([38,39]). For $\upsilon >0$, the classical digamma function $\psi \left(\upsilon \right)={\Gamma }^{\prime }\left(\upsilon \right)/\Gamma \left(\upsilon \right)$ satisfies

$$\log \upsilon -{\displaystyle \frac{1}{\upsilon }}<\psi \left(\upsilon \right)<\log \upsilon -{\displaystyle \frac{1}{2\upsilon }},\upsilon \in (0,\infty ).$$

(40)

1.3. A k-Digamma Inequality

Lemma 9. 

If $k>0$ and $\upsilon >0$, then the k-digamma function ${\psi }_k$ satisfies

$${\displaystyle \frac{1}{k}}\log \upsilon -{\displaystyle \frac{1}{\upsilon }}<{\psi }_k\left(\upsilon \right)<{\displaystyle \frac{1}{k}}\log \upsilon -{\displaystyle \frac{1}{2\upsilon }},\upsilon \in (0,\infty ).$$

(41)

Proof. 

From (16), we have

$$\log {\Gamma }_k\left(\upsilon \right)=\left({\displaystyle \frac{\upsilon }{k}}-1\right)\log k+\log \Gamma \left({\displaystyle \frac{\upsilon }{k}}\right).$$

(42)

Differentiating w.r.t $\upsilon$, we obtain

$${\displaystyle \frac{d}{d\upsilon }}\log {\Gamma }_k\left(\upsilon \right)={\displaystyle \frac{d}{d\upsilon }}\left[\left({\displaystyle \frac{\upsilon }{k}}-1\right)\log k\right]+{\displaystyle \frac{d}{d\upsilon }}\left[\log \Gamma \left({\displaystyle \frac{\upsilon }{k}}\right)\right].$$

(43)

After some simple calculation of (43), we get

$${\Gamma }_k^{\prime }\left(\upsilon \right)/{\Gamma }_k\left(\upsilon \right)={\displaystyle \frac{1}{k}}\log k+{\displaystyle \frac{1}{k}}\left({\Gamma }^{\prime }(\upsilon /k)/\Gamma (\upsilon /k)\right),$$

(44)

Finally, using the definition (17) and (18), we obtain

$${\psi }_k\left(\upsilon \right)={\displaystyle \frac{1}{k}}\log k+{\displaystyle \frac{1}{k}}\psi \left({\displaystyle \frac{\upsilon }{k}}\right),\upsilon \in (0,\infty ).$$

(45)

Using (40) with $\xi =\upsilon /k$,

$$\log \left({\displaystyle \frac{\upsilon }{k}}\right)-{\displaystyle \frac{k}{\upsilon }}<\psi \left({\displaystyle \frac{\upsilon }{k}}\right)<\log \left({\displaystyle \frac{\upsilon }{k}}\right)-{\displaystyle \frac{k}{2\upsilon }}.$$

(46)

Substituting (46) into (45), we get

$${\psi }_k\left(\upsilon \right)>{\displaystyle \frac{1}{k}}\left[\log \left({\displaystyle \frac{\upsilon }{k}}\right)-{\displaystyle \frac{k}{\upsilon }}+\log k\right]={\displaystyle \frac{1}{k}}\log \upsilon -{\displaystyle \frac{1}{\upsilon }},$$

(47)

and

$${\psi }_k\left(\upsilon \right)<{\displaystyle \frac{1}{k}}\left[\log \left({\displaystyle \frac{\upsilon }{k}}\right)-{\displaystyle \frac{k}{2\upsilon }}+\log k\right]={\displaystyle \frac{1}{k}}\log \upsilon -{\displaystyle \frac{1}{2\upsilon }}.$$

(48)

Combining (47) and (48), we obtain (41). □

2. Main Results

2.1. Starlikeness of the Normalized k-Mittag–Leffler Function

Theorem 1. 

Let $k>0$, $\sigma >0$, $\nu >0$ and $q>0$. Suppose that

$$k\le \sigma ,k\le \sigma \sqrt{{\displaystyle \frac{q}{\nu }}},k\nu \le \sqrt{2}\sigma q,$$

(49)

and

$$1+\log (q+k)-{\displaystyle \frac{k}{2(q+k)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma +\nu )+{\displaystyle \frac{\sigma }{\sigma +\nu }}<0.$$

If

$$2(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu ),$$

(50)

then ${\mathcal{E}}_{k,\sigma ,\nu }^q$ is starlike, ${\mathcal{E}}_{k,\sigma ,\nu }^q\in {\mathcal{S}}^{*}$.

Proof. 

To prove that ${\mathcal{E}}_{k,\sigma ,\nu }^q$ is starlike in D, it suffices to prove that

$$\left|{\displaystyle \frac{\tau {\left({\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\right)}^{\prime }}{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}}-1\right|<1,\tau \in D.$$

If

$${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)=\tau +{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}}{\tau }^{j+1},$$

then

$${\left({\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\right)}^{\prime }-{\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}{\tau }}={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{j{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}}{\tau }^j={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{B_j}{j!}}{\tau }^j,$$

(51)

where

$$B_j={\displaystyle \frac{j{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )}}.$$

(52)

Using ${\left(q\right)}_{j,k}={\displaystyle \frac{{\Gamma }_k(q+jk)}{{\Gamma }_k\left(q\right)}}$ into (52), we obtain

$$B_j={\displaystyle \frac{j{\Gamma }_k\left(\nu \right){\Gamma }_k(q+kj)}{{\Gamma }_k\left(q\right){\Gamma }_k(\sigma j+\nu )}}.$$

To show that $B_1\ge B_j,j=2,3,...,$ we consider

$$B\left(s\right)={\displaystyle \frac{s{\Gamma }_k\left(\nu \right){\Gamma }_k(q+ks)}{{\Gamma }_k\left(q\right){\Gamma }_k(\sigma s+\nu )}},s\ge 1.$$

Taking the logarithm and differentiating with respect to s, we get

$${\displaystyle \frac{B^{\prime }\left(s\right)}{B\left(s\right)}}={\displaystyle \frac{1}{s}}+k{\displaystyle \frac{{\Gamma }_k^{\prime }(q+ks)}{{\Gamma }_k(q+ks)}}-\sigma {\displaystyle \frac{{\Gamma }_k^{\prime }(\sigma s+\nu )}{{\Gamma }_k(\sigma s+\nu )}}={\displaystyle \frac{1}{s}}+k{\psi }_k(q+ks)-\sigma {\psi }_k(\sigma s+\nu ).$$

By Lemma 9, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{B^{\prime }\left(s\right)}{B\left(s\right)}}& <{\displaystyle \frac{1}{s}}+k\left({\displaystyle \frac{1}{k}}\log (q+ks)-{\displaystyle \frac{1}{2(q+ks)}}\right)-\sigma \left({\displaystyle \frac{1}{k}}\log (\sigma s+\nu )-{\displaystyle \frac{1}{\sigma s+\nu }}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{s}}+\log (q+ks)-{\displaystyle \frac{k}{2(q+ks)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma s+\nu )+{\displaystyle \frac{\sigma }{\sigma s+\nu }}.\hfill \end{array}$$

(53)

Therefore, we obtain

$${\displaystyle \frac{B^{\prime }\left(s\right)}{B\left(s\right)}}<\Omega \left(s\right),s\ge 1,$$

where

$$\Omega \left(s\right)={\displaystyle \frac{1}{s}}+\log (q+ks)-{\displaystyle \frac{k}{2(q+ks)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma s+\nu )+{\displaystyle \frac{\sigma }{\sigma s+\nu }}.$$

Now, we want to show that $\Omega \left(s\right)$ decreasing in $[1,\infty ):$

$$\begin{array}{cc}\hfill {\Omega }^{\prime }\left(s\right)& =-{\displaystyle \frac{1}{s^2}}+{\displaystyle \frac{k^2(\sigma s+\nu )-{\sigma }^2(q+ks)}{k(\sigma s+\nu )(q+ks)}}+{\displaystyle \frac{k^2{(\sigma s+\nu )}^2-2{\sigma }^2{(q+ks)}^2}{2{(q+ks)}^2{(\sigma s+\nu )}^2}}\hfill \\ \hfill & =-{\displaystyle \frac{1}{s^2}}+{\displaystyle \frac{(k^2\sigma -k{\sigma }^2)s+(k^2\nu -{\sigma }^{2}q)}{k(\sigma s+\nu )(q+ks)}}\hfill \\ \hfill & +{\displaystyle \frac{\left(k(\sigma s+\nu )-\sqrt{2}\sigma (q+ks)\right)\left(k(\sigma s+\nu )+\sqrt{2}\sigma (q+ks)\right)}{2{(q+ks)}^2{(\sigma s+\nu )}^2}}\hfill \\ \hfill & =-{\displaystyle \frac{1}{s^2}}+{\displaystyle \frac{(k^2\sigma -k{\sigma }^2)s+(k^2\nu -{\sigma }^{2}q)}{k(\sigma s+\nu )(q+ks)}}\hfill \\ \hfill & +{\displaystyle \frac{\left((k\sigma -\sqrt{2}\sigma k)s+(k\nu -\sqrt{2}\sigma q)\right)\left((k\sigma +\sqrt{2}\sigma k)s+(k\nu +\sqrt{2}\sigma q)\right)}{2{(q+ks)}^2{(\sigma s+\nu )}^2}}.\hfill \end{array}$$

(54)

Therefore, if

$$k\le \sigma ,k\le \sigma \sqrt{{\displaystyle \frac{q}{\nu }}},$$

and

$$k\nu \le \sqrt{2}\sigma q,$$

we have, ${\Omega }^{\prime }\left(s\right)<0,s\ge 1.$ Indeed, for every $s\ge 1$ the three terms in the last expression have the following signs. The first term $-1/s^2$ is negative. In the second term the denominator is positive, while $k^2\sigma -k{\sigma }^2=k\sigma (k-\sigma )\le 0$ because $k\le \sigma$, and $k^2\nu -{\sigma }^{2}q\le 0$ because $k\le \sigma \sqrt{q/\nu }$; hence its numerator is $\le 0$ for all $s\ge 1$. In the third term the denominator is positive, the second factor of the numerator is positive, and in the first factor $k\sigma -\sqrt{2}\sigma k=(1-\sqrt{2})k\sigma <0$ and $k\nu -\sqrt{2}\sigma q\le 0$; hence the first factor is negative for all $s\ge 1$. Consequently ${\Omega }^{\prime }\left(s\right)\le -1/s^2<0$ for all $s\ge 1$, and not only for large s. Note that the factor s in $B\left(s\right)$ contributes exactly the term $1/s$ to $B^{\prime }\left(s\right)/B\left(s\right)$, and the factors ${\Gamma }_k\left(\nu \right)$ and ${\Gamma }_k\left(q\right)$, being independent of s, contribute nothing. Thus $\Omega \left(s\right)$ is decreasing on $[1,\infty )$. In addition by hypothesis

$$\Omega \left(1\right)=1+\log (q+k)-{\displaystyle \frac{k}{2(q+k)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma +\nu )+{\displaystyle \frac{\sigma }{\sigma +\nu }}<0.$$

Since $\Omega$ is decreasing, it follows that

$$\Omega \left(s\right)<0,s\ge 1.$$

Consequently, we get

$${\displaystyle \frac{B^{\prime }\left(s\right)}{B\left(s\right)}}<0,s\ge 1.$$

Since $B\left(s\right)>0$, we obtain $B^{\prime }\left(s\right)<0,s\ge 1.$ Therefore, $B\left(s\right)$ is decreasing on $[1,\infty )$, and hence the sequence ${\left\{B_j\right\}}_{j\ge 1}$ is monotone decreasing. So we have

$$B_1={\displaystyle \frac{{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}\ge B_j,j\ge 1$$

and from (51) for $\tau \in D$,

$$\left|{\left({\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\right)}^{\prime }-{\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}{\tau }}\right|\le {\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{B_j}{j!}}{\left|\tau \right|}^j<{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{B_j}{j!}}\le B_1{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{1}{j!}}=B_1(e-1).$$

Thus

$$\left|{\left({\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\right)}^{\prime }-{\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}{\tau }}\right|<{\displaystyle \frac{{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}(e-1).$$

(55)

Next, we have

$$\left|{\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}{\tau }}\right|\ge 1-{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}}{\left|\tau \right|}^j,$$

where

$$C_j={\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )}}={\displaystyle \frac{B_j}{j}},$$

and ${\left\{B_j\right\}}_{j\ge 1}$ is decreasing, this follows that

$$C_j\le B_j\le B_1,j\ge 1.$$

Therefore,

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}{\tau }}\right|>1-{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{C_j}{j!}}\ge 1-B_1{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{1}{j!}}=1-B_1(e-1).\end{array}$$

Hence from (50), we have

$$\left|{\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}{\tau }}\right|>1-{\displaystyle \frac{{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}(e-1)>0.$$

(56)

Therefore, using (55) and (56), we obtain

$$\left|{\displaystyle \frac{\tau {\left({\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\right)}^{\prime }}{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}}-1\right|=\left|{\displaystyle \frac{{\left({\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\right)}^{\prime }-{\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}{\tau }}}{{\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}{\tau }}}}\right|<{\displaystyle \frac{{\displaystyle \frac{{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}(e-1)}{1-{\displaystyle \frac{{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}(e-1)}}.$$

(57)

This is equivalent to

$$2(e-1){\displaystyle \frac{{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}<1,$$

or

$$2(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu ),$$

which is exactly the hypothesis (50). Therefore, we obtain

$$\mathfrak{R}\left({\displaystyle \frac{\tau {\left({\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\right)}^{\prime }}{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}}\right)>0,\tau \in D.$$

Thus the function ${\mathcal{E}}_{k,\sigma ,\nu }^q$ is starlike in D. □

Corollary 1. 

If $q/\nu \le {\displaystyle \frac{1}{2}}$, then the three conditions in (49) reduce to the single condition $k\nu \le \sqrt{2}\sigma q$.

Proof. 

If $k\nu \le \sqrt{2}\sigma q$, then $k\le \sqrt{2}\sigma q/\nu =\sqrt{2q/\nu }\sigma \sqrt{q/\nu }\le \sigma \sqrt{q/\nu }\le \sigma /\sqrt{2}<\sigma$. □

Example 1. 

If $k=\nu =q=1\mathrm{and}\sigma =5$, then ${\mathcal{E}}_{k,\sigma ,\nu }^q$ is starlike.

Theorem 2. 

For $k>0$, $\sigma >0$, $\nu >0$ and $q>0$ assume that

$$k\le \sigma ,k\le \sigma \sqrt{{\displaystyle \frac{q}{\nu }}},k\nu \le \sqrt{2}\sigma q,$$

and

$$1+\log (q+k)-{\displaystyle \frac{k}{2(q+k)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma +\nu )+{\displaystyle \frac{\sigma }{\sigma +\nu }}<0.$$

If

$$3(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu ),$$

(58)

then the function ${\mathcal{E}}_{k,\sigma ,\nu }^q$ belongs to the class ${\mathcal{S}}_p$.

Proof. 

Proceeding similarly to the proof of Theorem 1 and applying part (ii) of Lemma 5 in (57), we obtain

$$3(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu ),$$

(59)

which is exactly condition (58). Therefore, by Lemma 5 (ii), it follows that ${\mathcal{E}}_{k,\sigma ,\nu }^q\in {\mathcal{S}}_p.$ This completes the proof. □

Corollary 2. 

If

$${\nu }_0<\nu \le 3\sqrt{2},$$

where ${\nu }_0$ is the positive root of

$$\nu (\nu +2)(\nu +4)=2(e-1),$$

then the function ${\mathcal{E}}_{2,6,\nu }^1\left(\tau \right)$ is starlike in D.

Proof. 

Taking $k=2,\sigma =6,q=1$ in Theorem 1, we get that ${\mathcal{E}}_{2,6,\nu }^1\left(\tau \right)$ is starlike in D if

$$2\le 6,2\le 6\sqrt{{\displaystyle \frac{1}{\nu }}},\nu \le 3\sqrt{2},$$

and

$$1+\log \left(3\right)-{\displaystyle \frac{1}{3}}-3\log (6+\nu )+{\displaystyle \frac{6}{6+\nu }}<0,$$

(60)

together with

$$2(e-1){\Gamma }_2\left(\nu \right)<{\Gamma }_2(\nu +6).$$

(61)

Hence the restrictions reduce to

$$0<\nu \le 3\sqrt{2}.$$

Using the recurrence relation (19), we have

$${\Gamma }_2(\nu +6)=(\nu +4){\Gamma }_2(\nu +4)=(\nu +4)(\nu +2){\Gamma }_2(\nu +2)=\nu (\nu +2)(\nu +4){\Gamma }_2\left(\nu \right).$$

(62)

Using (62) in (61) is equivalent to

$$2(e-1)<\nu (\nu +2)(\nu +4).$$

If

$$\varphi \left(\nu \right)=\nu (\nu +2)(\nu +4)-2(e-1),v>0,$$

then

$${\varphi }^{\prime }\left(\nu \right)=3{\nu }^2+12\nu +8>0,\nu >0.$$

Thus $\varphi$ is increasing on $(0,\infty )$. Let ${\nu }_0$ be the unique positive root of

$$\nu (\nu +2)(\nu +4)=2(e-1),$$

(numerically ${\nu }_0\approx 0.3387$), then $\varphi \left(\nu \right)>0$, whenever $\nu >{\nu }_0.$ Finally, define (60) as the function

$$h\left(\nu \right)=1+\log \left(3\right)-{\displaystyle \frac{1}{3}}-3\log (6+\nu )+{\displaystyle \frac{6}{6+\nu }},v>0.$$

Then

$$h^{\prime }\left(\nu \right)=-{\displaystyle \frac{3}{6+\nu }}-{\displaystyle \frac{6}{{(6+\nu )}^2}}<0,\nu >0.$$

Hence h is decreasing on $(0,\infty )$. Therefore, $h\left(\nu \right)<{\lim }_{\nu \to 0^{+}}h\left(\nu \right)={\displaystyle \frac{5}{3}}+\log 3-3\log 6\approx -2.61<0$ for every $\nu >0$, in particular for every ${\nu }_0<\nu \le 3\sqrt{2}.$ Consequently, all hypotheses of Theorem 1 are satisfied whenever

$${\nu }_0<\nu \le 3\sqrt{2}.$$

Hence, the function ${\mathcal{E}}_{2,6,\nu }^1\left(\tau \right)$ is starlike in D. □

Remark 3. 

Putting $k=1$ and $q=1$ in Theorems 1 and 2, we obtain starlikeness and parabolic starlikeness criteria for the normalized two-parameter ML function $\tau \Gamma \left(\nu \right)E_{\sigma ,\nu }\left(\tau \right)$; compare [12]. For $k=2$, $\sigma =6$, $q=1$, condition (58) is equivalent to $\nu (\nu +2)(\nu +4)>3(e-1)$, i.e. $\nu >{\nu }_2\approx 0.4676$. Hence, applying the above results gives: ${\mathcal{E}}_{2,6,\frac{2}{5}}^1\left(\tau \right)$ is starlike (since $\frac{2}{5}>{\nu }_0$), and ${\mathcal{E}}_{2,6,\frac{1}{2}}^1\left(\tau \right)\in {\mathcal{S}}_p$, ${\mathcal{E}}_{2,6,1}^1\left(\tau \right)\in {\mathcal{S}}_p$ (since ${\displaystyle \frac{1}{2}},1>{\nu }_2$).

Theorem 3. 

Let $k>0$, $\sigma >0$, $\nu >0$ and $q>0$. Suppose that

$$k\le \sigma ,k\le \sigma \sqrt{{\displaystyle \frac{q}{\nu }}},k\nu \le \sqrt{2}\sigma q,$$

and

$$\log (q+k)-{\displaystyle \frac{k}{2(q+k)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma +\nu )+{\displaystyle \frac{\sigma }{\sigma +\nu }}<0.$$

(63)

If

$$(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu ),$$

(64)

then the function ${\mathcal{E}}_{k,\sigma ,\nu }^q$ is starlike in

$$D_{1/2}=\left\{\tau :\left|\tau \right|<{\displaystyle \frac{1}{2}}\right\}.$$

Proof. 

It suffices to show that

$$\left|{\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}{\tau }}-1\right|<1,\tau \in D.$$

Then Lemma 2 implies that ${\mathcal{E}}_{k,\sigma ,\nu }^q$ is univalent and starlike in $D_{1/2}$. Recall the function

$${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)=\tau +{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}}{\tau }^{j+1}.$$

Thus

$${\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}{\tau }}-1={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}}{\tau }^j.$$

Put

$$C_j={\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )}},j\ge 1.$$

Define, for $s\ge 1$, the function

$$C\left(s\right)={\displaystyle \frac{{\Gamma }_k\left(\nu \right){\Gamma }_k(q+ks)}{{\Gamma }_k\left(q\right){\Gamma }_k(\sigma s+\nu )}},$$

then $C\left(j\right)=C_j.$ Taking logarithmic derivative, we obtain

$${\displaystyle \frac{C^{\prime }\left(s\right)}{C\left(s\right)}}=k{\psi }_k(q+ks)-\sigma {\psi }_k(\sigma s+\nu ).$$

By Lemma 9, we have

$${\displaystyle \frac{C^{\prime }\left(s\right)}{C\left(s\right)}}<\Lambda \left(s\right),$$

(65)

where

$$\Lambda \left(s\right)=\log (q+ks)-{\displaystyle \frac{k}{2(q+ks)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma s+\nu )+{\displaystyle \frac{\sigma }{\sigma s+\nu }},s\ge 1.$$

(66)

Now

$$\begin{array}{cc}\hfill {\Lambda }^{\prime }\left(s\right)& ={\displaystyle \frac{(k^2\sigma -k{\sigma }^2)s+(k^2\nu -{\sigma }^{2}q)}{k(\sigma s+\nu )(q+ks)}}\hfill \\ \hfill & +{\displaystyle \frac{\left((k\sigma -\sqrt{2}\sigma k)s+(k\nu -\sqrt{2}\sigma q)\right)\left((k\sigma +\sqrt{2}\sigma k)s+(k\nu +\sqrt{2}\sigma q)\right)}{2{(q+ks)}^2{(\sigma s+\nu )}^2}}.\hfill \end{array}$$

(67)

Under the given hypothesis, we have

$${\Lambda }^{\prime }\left(s\right)<0,s\ge 1.$$

Thus the function $\Lambda \left(s\right)$ is decreasing on $[1,\infty )$. Take s = 1 in (66), we have

$$\Lambda \left(1\right)=\log (q+k)-{\displaystyle \frac{k}{2(q+k)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma +\nu )+{\displaystyle \frac{\sigma }{\sigma +\nu }}.$$

By the hypothesis $\Lambda \left(1\right)<0$. Hence $\Lambda \left(s\right)<0,s\ge 1.$ Consequently (65) becomes

$${\displaystyle \frac{C^{\prime }\left(s\right)}{C\left(s\right)}}<0,s\ge 1.$$

Since $C\left(s\right)>0$, it follows that $C\left(s\right)$ is decreasing on $[1,\infty )$. Therefore,

$$C_j\le C_1,j\ge 1.$$

where

$$C_1={\displaystyle \frac{{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}.$$

Hence, for $\tau \in D$,

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)}{\tau }}-1\right|\le {\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{C_j}{j!}}{\left|\tau \right|}^j<{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{C_j}{j!}}\le C_1{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{1}{j!}}=C_1(e-1)={\displaystyle \frac{{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}(e-1)<1\end{array}$$

(68)

by hypothesis (64). Therefore, by Lemma 2, the function ${\mathcal{E}}_{k,\sigma ,\nu }^q$ is univalent and starlike in $D_{1/2}$. □

Example 2. 

The function ${\mathcal{E}}_{2,6,\frac{1}{3}}^1\left(\tau \right)$ is starlike in

$$D_{1/2}=\left\{\tau :\left|\tau \right|<{\displaystyle \frac{1}{2}}\right\}.$$

2.2. Convexity and Uniform Convexity of the Normalized k-Mittag–Leffler Function

Theorem 4. 

Let $k>0$, $\sigma >0$, $\nu >0$ and $q>0$. Suppose that

$$k\le \sigma ,k\le \sigma \sqrt{{\displaystyle \frac{q}{\nu }}},k\nu \le \sqrt{2}\sigma q,$$

and

$${\displaystyle \frac{3}{2}}+\log (q+k)-{\displaystyle \frac{k}{2(q+k)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma +\nu )+{\displaystyle \frac{\sigma }{\sigma +\nu }}<0.$$

If

$$4(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu ),$$

(69)

then the function ${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)$ is convex in D.

Proof. 

It is sufficient to prove that

$$G\left(\tau \right):=\tau {\left({\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\right)}^{\prime },\tau \in D$$

is starlike in D. Therefore

$$G\left(\tau \right)=\tau {\left({\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\right)}^{\prime }=\tau +{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{(j+1){\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}}{\tau }^{j+1}.$$

Also,

$$\begin{array}{c}\hfill G^{\prime }\left(\tau \right)-{\displaystyle \frac{G\left(\tau \right)}{\tau }}={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{j(j+1){\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}}{\tau }^j={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{D_j}{j!}}{\tau }^j,\end{array}$$

(70)

where

$$D_j={\displaystyle \frac{j(j+1){\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )}}.$$

We define, for $s\ge 1$, the function

$$D\left(s\right)={\displaystyle \frac{s(s+1){\Gamma }_k\left(\nu \right){\Gamma }_k(q+ks)}{{\Gamma }_k\left(q\right){\Gamma }_k(\sigma s+\nu )}},$$

then $D\left(j\right)=D_j,j\ge 1.$ Taking the logarithmic differentiation, we obtain

$${\displaystyle \frac{D^{\prime }\left(s\right)}{D\left(s\right)}}={\displaystyle \frac{1}{s}}+{\displaystyle \frac{1}{s+1}}+k{\psi }_k(q+ks)-\sigma {\psi }_k(\sigma s+\nu ).$$

By Lemma 9, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{D^{\prime }\left(s\right)}{D\left(s\right)}}<\Theta (s),s\ge 1,\end{array}$$

(71)

where

$$\Theta \left(s\right)={\displaystyle \frac{1}{s}}+{\displaystyle \frac{1}{s+1}}+\log (q+ks)-{\displaystyle \frac{k}{2(q+ks)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma s+\nu )+{\displaystyle \frac{\sigma }{\sigma s+\nu }}.$$

Taking the derivative, we get

$$\begin{array}{cc}\hfill {\Theta }^{\prime }\left(s\right)& =-{\displaystyle \frac{1}{s^2}}-{\displaystyle \frac{1}{{(s+1)}^2}}+{\displaystyle \frac{(k^2\sigma -k{\sigma }^2)s+(k^2\nu -{\sigma }^{2}q)}{k(\sigma s+\nu )(q+ks)}}\hfill \\ \hfill & +{\displaystyle \frac{\left((k\sigma -\sqrt{2}\sigma k)s+(k\nu -\sqrt{2}\sigma q)\right)\left((k\sigma +\sqrt{2}\sigma k)s+(k\nu +\sqrt{2}\sigma q)\right)}{2{(q+ks)}^2{(\sigma s+\nu )}^2}}.\hfill \end{array}$$

(72)

By hypothesis, we see that

$${\Theta }^{\prime }\left(s\right)<0,s\ge 1.$$

Thus $\Theta \left(s\right)$ is decreasing on $[1,\infty )$. By the hypothesis,

$$\Theta \left(1\right)={\displaystyle \frac{3}{2}}+\log (q+k)-{\displaystyle \frac{k}{2(q+k)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma +\nu )+{\displaystyle \frac{\sigma }{\sigma +\nu }}<0.$$

Since $\Theta$ is decreasing, it follows that $\Theta \left(s\right)<0,s\ge 1.$ Consequently, ${\displaystyle \frac{D^{\prime }\left(s\right)}{D\left(s\right)}}<0,s\ge 1.$ Since $D\left(s\right)>0$, we obtain $D^{\prime }\left(s\right)<0,s\ge 1.$ Therefore $D\left(s\right)$ is decreasing on $[1,\infty )$, and hence the sequence ${\left\{D_j\right\}}_{j\ge 1}$ is monotone decreasing. Hence $D_j\le D_1,j\ge 1.$ Since

$$D_1={\displaystyle \frac{2{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}},$$

we get, for $\tau \in D$,

$$\begin{array}{c}\hfill \left|G^{\prime }\left(\tau \right)-{\displaystyle \frac{G\left(\tau \right)}{\tau }}\right|\le {\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{D_j}{j!}}{\left|\tau \right|}^j<{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{D_j}{j!}}\le D_1{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{1}{j!}}=D_1(e-1)={\displaystyle \frac{2{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}(e-1).\end{array}$$

Next, we estimate $\left|{\displaystyle \frac{G\left(\tau \right)}{\tau }}\right|$, we have

$$\left|{\displaystyle \frac{G\left(\tau \right)}{\tau }}\right|\ge 1-{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{(j+1){\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}}{\left|\tau \right|}^j.$$

Let

$$E_j={\displaystyle \frac{(j+1){\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )}}.$$

Since $E_j={\displaystyle \frac{D_j}{j}},$ and ${\left\{D_j\right\}}_{j\ge 1}$ is decreasing, it follows that

$$E_j\le D_j\le D_1,j\ge 1.$$

Therefore, for $\tau \in D$

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{G\left(\tau \right)}{\tau }}\right|>1-{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{E_j}{j!}}\ge 1-D_1{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{1}{j!}}=1-D_1(e-1)=1-{\displaystyle \frac{2{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}(e-1)>0.\end{array}$$

(73)

by (69). Hence for $\tau \in D$

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{\tau G^{\prime }\left(\tau \right)}{G\left(\tau \right)}}-1\right|& =\left|{\displaystyle \frac{G^{\prime }\left(\tau \right)-{\displaystyle \frac{G\left(\tau \right)}{\tau }}}{{\displaystyle \frac{G\left(\tau \right)}{\tau }}}}\right|\hfill \\ \hfill & <{\displaystyle \frac{{\displaystyle \frac{2{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}(e-1)}{1-{\displaystyle \frac{2{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}(e-1)}}<1.\hfill \end{array}$$

(74)

This is equivalent to

$$4(e-1){\displaystyle \frac{{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}<1,$$

or

$$4(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu ),$$

which is exactly the hypothesis (69). Thus G is starlike in D. Since

$$G\left(\tau \right)=\tau {\left({\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\right)}^{\prime },$$

it follows that the function ${\mathcal{E}}_{k,\sigma ,\nu }^q$ is convex in D. □

Example 3. 

The function ${\mathcal{E}}_{2,6,1}^1\left(\tau \right)$ is convex in D.

Theorem 5. 

Let $k>0$, $\sigma >0$, $\nu >0$ and $q>0$. Suppose that

$$k\le \sigma ,k\le \sigma \sqrt{{\displaystyle \frac{q}{\nu }}},k\nu \le \sqrt{2}\sigma q,$$

and

$${\displaystyle \frac{1}{2}}+\log (q+k)-{\displaystyle \frac{k}{2(q+k)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma +\nu )+{\displaystyle \frac{\sigma }{\sigma +\nu }}<0.$$

Then the following assertions hold:

(i) 

If

$$2(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu ),$$

then the function ${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)$ is convex in

$$D_{1/2}=\left\{\tau :\left|\tau \right|<{\displaystyle \frac{1}{2}}\right\}.$$

(ii) 

If

$$\sqrt{5}(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu ),$$

then the function ${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)$ is starlike in D.

Proof. 

Write $\chi ={\mathcal{E}}_{k,\sigma ,\nu }^q$. By (28),

$${\chi }^{\prime }\left(\tau \right)-1={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{E_j}{j!}}{\tau }^j,E_j={\displaystyle \frac{(j+1){\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )}}.$$

For $s\ge 1$ put

$$\tilde{E}\left(s\right)={\displaystyle \frac{(s+1){\Gamma }_k\left(\nu \right){\Gamma }_k(q+ks)}{{\Gamma }_k\left(q\right){\Gamma }_k(\sigma s+\nu )}},$$

so that $\tilde{E}\left(j\right)=E_j$. By Lemma 9 and (66),

$${\displaystyle \frac{{\tilde{E}}^{\prime }\left(s\right)}{\tilde{E}\left(s\right)}}={\displaystyle \frac{1}{s+1}}+k{\psi }_k(q+ks)-\sigma {\psi }_k(\sigma s+\nu )<{\displaystyle \frac{1}{s+1}}+\Lambda \left(s\right)=:\Phi \left(s\right).$$

From the expression of ${\Lambda }^{\prime }\left(s\right)$ in the proof of Theorem 3, ${\Phi }^{\prime }\left(s\right)=-{\displaystyle \frac{1}{{(s+1)}^2}}+{\Lambda }^{\prime }\left(s\right)<0$ under the stated parameter conditions, and by hypothesis $\Phi \left(1\right)={\displaystyle \frac{1}{2}}+\Lambda \left(1\right)<0$. Hence $\Phi \left(s\right)<0$, so $\tilde{E}$ is decreasing on $[1,\infty )$ and

$$E_j\le E_1={\displaystyle \frac{2{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}},j\ge 1.$$

Therefore, for $\tau \in D$,

$$|{\chi }^{\prime }\left(\tau \right)-1|<E_1{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{1}{j!}}={\displaystyle \frac{2(e-1){\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}.$$

(i) If $2(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu )$, then $|{\chi }^{\prime }\left(\tau \right)-1|<1$ in D, and Lemma 4 gives convexity in $D_{1/2}$.

(ii) If $\sqrt{5}(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu )$, then $|{\chi }^{\prime }\left(\tau \right)-1|<2/\sqrt{5}$ in D, and Lemma 3 gives $\chi \in {\mathcal{S}}^{*}$. □

Remark 4. 

Part (ii) of Theorem 5 requires a stronger gamma condition than Theorem 1 ($\sqrt{5}>2$) but a weaker logarithmic condition (${\displaystyle \frac{1}{2}}$ instead of 1); hence the two results are independent.

Theorem 6. 

Let $k>0$, $\sigma >0$, $\nu >0$ and $q>0$. Suppose that

$$k\le \sigma ,k\le \sigma \sqrt{{\displaystyle \frac{q}{\nu }}},k\nu \le \sqrt{2}\sigma q,$$

and

$${\displaystyle \frac{3}{2}}+\log (q+k)-{\displaystyle \frac{k}{2(q+k)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma +\nu )+{\displaystyle \frac{\sigma }{\sigma +\nu }}<0.$$

If

$$6(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu ),$$

(75)

then the function ${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)$ is uniformly convex in D.

Proof. 

By Lemma 5 (i), it is sufficient to prove that

$$\left|{\displaystyle \frac{\tau {\left({\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\right)}^{″}}{{\left({\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\right)}^{\prime }}}\right|<{\displaystyle \frac{1}{2}},\tau \in D.$$

With $\chi ={\mathcal{E}}_{k,\sigma ,\nu }^q$ and $D_j$, $E_j$ as in the proof of Theorem 4,

$$\tau {\chi }^{″}\left(\tau \right)={\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{D_j}{j!}}{\tau }^j,{\chi }^{\prime }\left(\tau \right)=1+{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{E_j}{j!}}{\tau }^j.$$

Under the present hypotheses (which are those of Theorem 4), the proof of Theorem 4 gives $D_j\le D_1=2{\Gamma }_k\left(\nu \right)q/{\Gamma }_k(\sigma +\nu )$ and $E_j=D_j/j\le D_1$ for $j\ge 1$. Hence, for $\tau \in D$,

$$\left|{\displaystyle \frac{\tau {\chi }^{″}\left(\tau \right)}{{\chi }^{\prime }\left(\tau \right)}}\right|<{\displaystyle \frac{(e-1)D_1}{1-(e-1)D_1}}<{\displaystyle \frac{1}{2}}⟺3(e-1)D_1<1⟺6(e-1){\Gamma }_k\left(\nu \right)q<{\Gamma }_k(\sigma +\nu ),$$

which is (75). By Lemma 5(i), it follows that

$${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)\in \mathcal{UCV}.$$

□

Example 4. 

The function ${\mathcal{E}}_{2,6,2}^1\left(\tau \right)$ is uniformly convex in D.

Proof. 

For $k=2,\sigma =6,\nu =2,q=1.$ All hypotheses of Theorem 6 are satisfied. Hence, ${\mathcal{E}}_{2,6,2}^1\left(\tau \right)\in \mathcal{UCV}.$ □

2.3. Close-to-Convexity of the Normalized k-Mittag–Leffler Function with Respect to Starlike Functions

Theorem 7. 

Let $k>0$, $\sigma >0$, $\nu >0$ and $q>0$. Suppose that

$$k\le \sigma ,k\le \sigma \sqrt{{\displaystyle \frac{q}{\nu }}},k\nu \le \sqrt{2}\sigma q,$$

and

$$1+\log (q+k)-{\displaystyle \frac{k}{2(q+k)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma +\nu )+{\displaystyle \frac{\sigma }{\sigma +\nu }}<0.$$

If

$$2{\Gamma }_k\left(\nu \right)q\le {\Gamma }_k(\sigma +\nu ),$$

then the function ${\mathcal{E}}_{k,\sigma ,\nu }^q$ is close-to-convex with respect to $-\log (1-\tau )$ in D.

Proof. 

We have

$${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)=\tau +{\displaystyle \sum _{l=2}^{\infty }}{a}_l{\tau }^l,$$

where

$$a_l={\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{l-1,k}}{{\Gamma }_k(\sigma (l-1)+\nu )(l-1)!}},l\ge 2.$$

By Lemma 6, it is enough to prove that

$$1\ge 2a_2\ge 3a_3\ge \cdots \ge l{a}_l\ge (l+1)a_{l+1}\ge \cdots \ge 0.$$

Since all parameters are positive, we have $a_l>0,l\ge 2.$ Now define

$$B_j={\displaystyle \frac{j{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )}},j\ge 1.$$

From the proof of Theorem 1, under the present hypotheses, the sequence ${\left\{B_j\right\}}_{j\ge 1}$ is monotone decreasing. Hence $B_{j+1}\le B_j,j\ge 1.$ Also,

$$(j+1)a_{j+1}={\displaystyle \frac{(j+1){\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}}={\displaystyle \frac{(j+1)B_j}{jj!}},j\ge 1.$$

Therefore,

$${\displaystyle \frac{(j+2)a_{j+2}}{(j+1)a_{j+1}}}={\displaystyle \frac{(j+2)B_{j+1}}{(j+1)(j+1)!}}·{\displaystyle \frac{jj!}{(j+1)B_j}}={\displaystyle \frac{j(j+2)}{{(j+1)}^3}}·{\displaystyle \frac{B_{j+1}}{B_j}}.$$

Since $B_{j+1}\le B_j$ and $j(j+2)={(j+1)}^2-1<{(j+1)}^3$, we get

$${\displaystyle \frac{(j+2)a_{j+2}}{(j+1)a_{j+1}}}\le {\displaystyle \frac{j(j+2)}{{(j+1)}^3}}<1,j\ge 1.$$

Thus

$$(j+2)a_{j+2}\le (j+1)a_{j+1},j\ge 1.$$

Hence

$$2a_2\ge 3a_3\ge \cdots \ge l{a}_l\ge (l+1)a_{l+1}\ge \cdots \ge 0.$$

Finally,

$$a_2={\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{1,k}}{{\Gamma }_k(\sigma +\nu )}}={\displaystyle \frac{{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}.$$

Hence

$$2a_2={\displaystyle \frac{2{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}.$$

By the hypothesis

$$2{\Gamma }_k\left(\nu \right)q\le {\Gamma }_k(\sigma +\nu ),$$

we obtain $2a_2\le 1.$ Therefore, by Lemma 6, the function ${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)$ is close-to-convex relative to $-\log (1-\tau )$ in D. □

Corollary 3. 

Let ${\nu }_1\approx 0.2143$ be the unique positive root of $\nu (\nu +2)(\nu +4)=2$. If ${\nu }_1\le \nu \le 3\sqrt{2}$, then the function ${\mathcal{E}}_{2,6,\nu }^1\left(\tau \right)$ is close-to-convex in D with respect to $-\log (1-\tau )$.

Proof. 

Put $k=2,\sigma =6,q=1$ in Theorem 7, and then $\nu \le 9.$ Moreover, $\nu \le 3\sqrt{2}.$ Thus the parameter restrictions are satisfied whenever $0<\nu \le 3\sqrt{2}.$ Now the logarithmic condition becomes

$$1+\log 3-{\displaystyle \frac{1}{3}}-3\log (6+\nu )+{\displaystyle \frac{6}{6+\nu }}<0.$$

This holds for every $\nu >0$ (see the proof of Corollary 2). Finally, by (62), the condition

$$2{\Gamma }_2\left(\nu \right)\le {\Gamma }_2(\nu +6)$$

is equivalent to $\nu (\nu +2)(\nu +4)\ge 2$, i.e., to $\nu \ge {\nu }_1$, since the left-hand side is increasing in $\nu$. This gives that all hypotheses of Theorem 7 are satisfied. Thus, the function ${\mathcal{E}}_{2,6,\nu }^1\left(\tau \right)$ is close-to-convex with respect to $-\log (1-\tau )$ in D. □

Corollary 4. 

The function ${\mathcal{E}}_{2,6,2}^1\left(\tau \right)$ is close-to-convex in D with respect to the function $-\log (1-\tau )$.

Remark 5. 

Theorem 7 extends the corresponding close-to-convexity result for the classical normalized ML function to the normalized k-ML function. Owing to the presence of the additional parameter $k>0$, the normalized k-ML function forms a wider class than its classical counterpart. Consequently, Theorem 7 provides a more general sufficient condition for close-to-convexity in D with respect to the starlike function $-\log (1-\tau )$. In particular, when $k=1$, the normalized k-ML function reduces to the normalized Prabhakar function (and for $q=1$ to the normalized two-parameter ML function), and the corresponding close-to-convexity result is recovered.

Theorem 8. 

Let $k>0$, $\sigma >0$, $\nu >0$ and $q>0$. Suppose that

$$k\le \sigma ,k\le \sigma \sqrt{{\displaystyle \frac{q}{\nu }}},k\nu \le \sqrt{2}\sigma q,$$

and

$$1+\log (q+k)-{\displaystyle \frac{k}{2(q+k)}}-{\displaystyle \frac{\sigma }{k}}\log (\sigma +\nu )+{\displaystyle \frac{\sigma }{\sigma +\nu }}<0.$$

If

$$3{\Gamma }_k\left(\nu \right)q\le {\Gamma }_k(\sigma +\nu ),$$

then the odd function

$${\mathbb{E}}_{k,\sigma ,\nu }^q\left(\tau \right)={\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left({\tau }^2\right)}{\tau }}={\Gamma }_k\left(\nu \right)\tau E_{k,\sigma ,\nu }^q\left({\tau }^2\right)$$

is close-to-convex with respect to

$${\displaystyle \frac{1}{2}}\log \left({\displaystyle \frac{1+\tau }{1-\tau }}\right)$$

in D.

Proof. 

If

$${\mathcal{E}}_{k,\sigma ,\nu }^q\left(\tau \right)=\tau +{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}}{\tau }^{j+1},$$

and

$${\mathbb{E}}_{k,\sigma ,\nu }^q\left(\tau \right)={\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left({\tau }^2\right)}{\tau }}.$$

Then

$${\mathbb{E}}_{k,\sigma ,\nu }^q\left(\tau \right)=\tau +{\displaystyle \sum _{j=1}^{\infty }}{\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}}{\tau }^{2j+1}.$$

Let

$$A_{2j+1}={\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}},j\ge 1.$$

Then

$${\mathbb{E}}_{k,\sigma ,\nu }^q\left(\tau \right)=\tau +{\displaystyle \sum _{j=1}^{\infty }}{A}_{2j+1}{\tau }^{2j+1}.$$

By Lemma 7, it is enough to prove that

$$1\ge 3A_3\ge 5A_5\ge \cdots \ge (2j+1)A_{2j+1}\ge (2j+3)A_{2j+3}\ge \cdots \ge 0.$$

Since all parameters are positive, we have

$$A_{2j+1}>0,j\ge 1.$$

Now define

$$B_j={\displaystyle \frac{j{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )}},j\ge 1.$$

From the proof of Theorem 1, under the present hypotheses, the sequence ${\left\{B_j\right\}}_{j\ge 1}$ is monotonically decreasing. Hence $B_{j+1}\le B_j,j\ge 1.$ Also,

$$A_{2j+1}={\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{j,k}}{{\Gamma }_k(\sigma j+\nu )j!}}={\displaystyle \frac{B_j}{jj!}},j\ge 1.$$

Therefore,

$$(2j+1)A_{2j+1}={\displaystyle \frac{(2j+1)B_j}{jj!}},j\ge 1.$$

Similarly,

$$(2j+3)A_{2j+3}={\displaystyle \frac{(2j+3)B_{j+1}}{(j+1)(j+1)!}}.$$

Hence

$$\begin{array}{cc}\hfill {\displaystyle \frac{(2j+3)A_{2j+3}}{(2j+1)A_{2j+1}}}& ={\displaystyle \frac{{\displaystyle \frac{(2j+3)B_{j+1}}{(j+1)(j+1)!}}}{{\displaystyle \frac{(2j+1)B_j}{jj!}}}}\hfill \\ \hfill & ={\displaystyle \frac{j(2j+3)B_{j+1}j!}{(j+1)(2j+1)B_j(j+1)!}}\hfill \\ \hfill & ={\displaystyle \frac{j(2j+3)B_{j+1}}{{(j+1)}^2(2j+1)B_j}}.\hfill \end{array}$$

(76)

Since $B_{j+1}\le B_j$, we get

$${\displaystyle \frac{(2j+3)A_{2j+3}}{(2j+1)A_{2j+1}}}\le {\displaystyle \frac{j(2j+3)}{{(j+1)}^2(2j+1)}}.$$

Now

$$j(2j+3)=2j^2+3j$$

and

$${(j+1)}^2(2j+1)=(j^2+2j+1)(2j+1)=2j^3+5j^2+4j+1.$$

Clearly,

$$2j^2+3j\le 2j^3+5j^2+4j+1,j\ge 1.$$

Thus

$${\displaystyle \frac{(2j+3)A_{2j+3}}{(2j+1)A_{2j+1}}}\le 1.$$

Hence

$$(2j+3)A_{2j+3}\le (2j+1)A_{2j+1},j\ge 1.$$

Therefore,

$$3A_3\ge 5A_5\ge \cdots \ge (2j+1)A_{2j+1}\ge (2j+3)A_{2j+3}\ge \cdots \ge 0.$$

It remains to show that

$$1\ge 3A_3.$$

For $j=1$, we have

$$A_3={\displaystyle \frac{{\Gamma }_k\left(\nu \right){\left(q\right)}_{1,k}}{{\Gamma }_k(\sigma +\nu )}}={\displaystyle \frac{{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}.$$

Hence

$$3A_3={\displaystyle \frac{3{\Gamma }_k\left(\nu \right)q}{{\Gamma }_k(\sigma +\nu )}}.$$

By the hypothesis

$$3{\Gamma }_k\left(\nu \right)q\le {\Gamma }_k(\sigma +\nu ),$$

we get

$$3A_3\le 1.$$

Consequently,

$$1\ge 3A_3\ge 5A_5\ge \cdots \ge (2j+1)A_{2j+1}\ge (2j+3)A_{2j+3}\ge \cdots \ge 0.$$

Therefore, by Lemma 7, the function

$${\mathbb{E}}_{k,\sigma ,\nu }^q\left(\tau \right)={\displaystyle \frac{{\mathcal{E}}_{k,\sigma ,\nu }^q\left({\tau }^2\right)}{\tau }}$$

is close-to-convex with respect to

$${\displaystyle \frac{1}{2}}\log \left({\displaystyle \frac{1+\tau }{1-\tau }}\right)$$

in D. □

2.4. Conclusions and Future Directions

In this paper, we have investigated several geometric properties of a normalized k-Mittag–Leffler function associated with the k-Gamma function and the k-Pochhammer symbol. Unlike the classical Mittag–Leffler and Le Roy-type Mittag–Leffler functions, the coefficients of the normalized k-Mittag–Leffler function involve both the k-Gamma function and the k-Pochhammer symbol, making the existing coefficient techniques inapplicable. To overcome this difficulty, we first established a new two-sided inequality for the k-digamma function and employed it to derive monotonicity properties of the corresponding coefficient sequences. These auxiliary results formed the principal analytical tools for our investigation.

By combining the obtained monotonicity results with several classical lemmas from geometric function theory, we derived explicit sufficient conditions for the normalized k-Mittag–Leffler function to be starlike, convex, uniformly convex, parabolic starlike, and close-to-convex in the unit disk. Corresponding starlikeness and convexity results in the half-unit disk were also established. Furthermore, we investigated the associated odd normalized function and obtained sufficient conditions for its close-to-convexity with respect to the function

$${\displaystyle \frac{1}{2}}\log \left({\displaystyle \frac{1+\tau }{1-\tau }}\right).$$

Several corollaries and illustrative examples were presented to demonstrate the applicability of the obtained criteria. In the case $k=1$, our results reduce to geometric properties of the normalized Prabhakar function and, for $k=q=1$, of the normalized two-parameter Mittag–Leffler function. Consequently, the present work carries the coefficient-monotonicity method from the classical Gamma-function framework over to the k-Gamma setting.

The techniques developed in this paper suggest several promising directions for future research. One possible extension is the investigation of geometric properties of more general normalized Mittag–Leffler-type functions involving additional parameters or other generalized Gamma functions. Another interesting direction is the study of subclasses associated with differential subordinations, superordinations, q-calculus, fractional integral and differential operators, and integral transforms acting on the normalized k-ML function. It would also be worthwhile to investigate coefficient estimates, Hankel and Toeplitz determinants, logarithmic coefficients, Fekete–Szegö inequalities, radius problems, and Bohr-type inequalities for this class. Finally, since ML-type functions play an important role in fractional differential equations and mathematical physics, the normalized k-ML function considered here may also find applications in fractional calculus, generalized fractional operators, and other models arising in applied mathematics.