Partial sums of generalized <em>q</em>-Mittag-Leffler functions

Muhammad Sabil Ur Rehman; Qazi Zahoor Ahmad; Hari M. Srivastava; Bilal Khan; Nazar Khan; Muhammad Sabil Ur Rehman; Qazi Zahoor Ahmad; Hari M. Srivastava; Bilal Khan; Nazar Khan

doi:10.3934/math.2020028

AIMS Mathematics

2020, Volume 5, Issue 1: 408-420. doi: 10.3934/math.2020028

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Research article

Partial sums of generalized q-Mittag-Leffler functions

1.
Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad, Pakistan
2.
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China
4.
Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan
5.
School of Mathematical Sciences, East China Normal University, 500 Dongchuan Road, Shanghai 200241, People’s Republic of China

Received: 04 October 2019 Accepted: 25 November 2019 Published: 02 December 2019
MSC : Primary 05A30, 30C45; Secondary 11B65, 47B38

In the present investigation, our main aim is to give lower bounds for the ratio of some normalized q-Mittag-Leffler function and their sequences of partial sums. We consider various corollaries and consequences of our main results.

Keywords:

univalent functions,
analytic functions,
partial sums,
q-derivative (or q-difference) operator,
normalized q-Mittag-Leffler function

Citation: Muhammad Sabil Ur Rehman, Qazi Zahoor Ahmad, Hari M. Srivastava, Bilal Khan, Nazar Khan. Partial sums of generalized q-Mittag-Leffler functions[J]. AIMS Mathematics, 2020, 5(1): 408-420. doi: 10.3934/math.2020028

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Abstract

1. Introduction

Let $\mathcal{A}$ denote the class of all functions $f$ which are analytic in the open unit disk

$\begin{equation*} \mathbb{U} = \left\{ z:z\in \mathbb{C} \ \ \ \text{and }\ \ \ \left\vert z\right\vert \lt 1\right\} \end{equation*}$

and normalized by the following condition:

$\begin{equation*} f\left( 0\right) = 0 = f^{\prime }\left( 0\right) -1, \end{equation*}$

that is, a function $f\in \mathcal{A}$ has the following Taylor-Maclaurin series representation:

$\begin{equation} f\left( z\right) = z+\sum\limits_{n = 2}^{\infty }a_{n}z^{n}\ \ \ \ \ \ \ \ \ \left( z\in \mathbb{U}\right) . \end{equation}$

(1.1)

Let $\mathcal{S}$ be the subclass of $\mathcal{A}$ consisting of all univalent functions in $\mathbb{U}$ .

We denote the class of starlike functions by $\mathcal{S}^{\ast}$ , which is the usual subclass of the normalized univalent function class $\mathcal{S}$ . That is, $\mathcal{S}^{\ast }$ consists of functions $f\in \mathcal{A}$ that satisfy the following inequality:

$\begin{equation*} \Re \left( \frac{zf^{\prime }\left( z\right) }{f\left( z\right) }\right) \gt 0\ \ \ \ \ \ \ \ \ \left( z\in \mathbb{U}\right) . \end{equation*}$

We now recall some basic definitions and concept details of the $q$ -calculus, which are used in this paper. We suppose throughout the paper that $0 < q < 1$ and that

$\begin{equation*} \mathbb{N} = \left\{ 1,2,3,...\right\} = \mathbb{N} _{0}\setminus \left\{ 0\right\} \ \ \ \ \ \ \ \left( \mathbb{N} _{0} = \left\{ 0,1,2,...\right\} \right) . \end{equation*}$

Definition 1.1. Let $q\in \left(0, 1\right)$ and define the $q$ -number $\left[\lambda \right] _{q}$ by

$\begin{equation*} \left[ \lambda \right] _{q} = \left\{ \begin{array}{ll} \dfrac{1-q^{\lambda }}{1-q} & \left( \lambda \in \mathbb{C} \right) \\ & \\ \sum\limits_{k = 0}^{n-1}q^{k} = 1+q+q^{2}+...+q^{n-1} & \left( \lambda = n\in \mathbb{N} \right) . \end{array} \right. \end{equation*}$

Definition 1.2. Let $q\in \left(0, 1\right)$ and define the $q$ -factorial $\left[ n\right] _{q}!$ by

$\begin{equation*} \left[ n\right] _{q}! = \left\{ \begin{array}{ll} 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & n = 0 \\ & \\ \prod\limits_{k = 1}^{n}\left[ k\right] _{q} & n\in \mathbb{N} . \end{array} \right. \end{equation*}$

Definition 1.3. Let $q\in \left(0, 1\right)$ and define $q$ -generalized Pochhammer symbol by

$\begin{equation*} \left( \left[ t\right] _{q}\right) _{n} = \left\{ \begin{array}{ll} 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & (n = 0) \\ & \\ \prod\limits_{k = 0}^{n}\left[ t+k\right] _{q} & (n\in \mathbb{N}) . \end{array} \right. \end{equation*}$

We note that

$\begin{equation} \left( \left[ t\right] _{q}\right) _{n} = \left[ t\right] _{q}\left( \left[ t+1 \right]_{q}\right)_{n-1}\ \ \ \ \ \ \ \ \ \ \ (n\in \mathbb{N}) \end{equation}$

(1.2)

and

$\begin{equation} \left( \left[ t\right] _{q}\right) _{n}\geqq \left( \left[ t\right] _{q}\right)^{n} \ \ \ \ \ \ \ \ (n\in \mathbb{N}) . \end{equation}$

(1.3)

Definition 1.4. For $t > 0$ , let the $q$ -gamma function be defined by

$\begin{equation*} \Gamma _{q}\left( t+1\right) = \left[ t\right] _{q}\Gamma _{q}\left( t\right) \ \ \ \ \ \text{and }\ \ \ \ \ \Gamma _{q}\left( 1\right) = 1. \end{equation*}$

Definition 1.5. (see ^[9] and ^[10]; see also ^[1,20] and ^[27]) The $q$ -derivative (or the $q$ -difference) operator $D_{q}$ for a function $f\in \mathcal{A}$ in given subset of $\mathbb{C}$ is defined by

$\begin{equation} D_{q}f\left( z\right) = \left\{ \begin{array}{ll} \dfrac{f\left( z\right) -f\left( qz\right) }{\left( 1-q\right) z}\ \ \ \ \ \ \ \ \ \ & \left(z\not = 0\right) \\ & \\ f^{\prime }\left( 0\right) & \left(z = 0\right), \end{array} \right. \end{equation}$

(1.4)

provided that $f^{\prime}\left(0\right)$ exists.

We deduce from Definition 1.5 that

$\begin{equation*} \lim\limits_{q\rightarrow 1^{-}}\left( D_{q}f\right) \left( z\right) = \lim\limits_{q\rightarrow 1-}\left( \frac{f\left( z\right) -f\left( qz\right) }{ \left( 1-q\right) z}\right) = f^{\prime }\left( z\right) \end{equation*}$

for a differentiable function $f$ in a given subset of $\mathbb{C}.$ It can be easily seen from $\left(1.1\right)$ and (1.4) that

$\begin{equation} \left( D_{q}f\right) \left( z\right) = 1+\sum\limits_{n = 2}^{\infty }\left[ n\right] _{q}a_{n}z^{n-1}. \end{equation}$

(1.5)

In geometric function theory, the operator $D_{q}$ $\left(\text{see Definition 1.5}\right)$ provides an important tool that has been used in order to investigate various subclasses of the class $\mathcal{S}$ of normalized univalent functions. Historically speaking, Ismail et al. (see ^[8]) were the first who introduced a $q$ -analogue of the class $\mathcal{S}^{\ast}$ of normalized starlike functions in $\mathbb{U}$ $\left(\text{see Definition 1.6 below}\right)$ . However, an important usage of the $q$ -calculus in the context of geometric function theory was actually provided and the basic (or $q$ -) hypergeometric functions were first used in geometric function theory in a book chapter by Srivastava (see, for details, [22,pp. 347 et seq.] (see also some more recent works ^[13,24].

Definition 1.6. (see ^[8] and ^[27]) A function $f\in \mathcal{A}$ is said to belong to the class $\mathcal{S}_{q}^{\ast }$ if

$\begin{equation} f\left( 0\right) = 0 = f^{\prime }\left( 0\right) -1 \end{equation}$

(1.6)

and

$\begin{equation} \left\vert \frac{z}{f\left( z\right) }\left( D_{q}f\right) \left( z\right)- \frac{1}{1-q}\right\vert \leqq \frac{1}{1-q} \ \ \ \ \ \ \ \ \ (z\in \mathbb{U}). \end{equation}$

(1.7)

It is readily observed that, as $q\rightarrow 1-$ , the closed disk

$\begin{equation*} \left\vert w-\frac{1}{1-q}\right\vert \leqq \frac{1}{1-q} \end{equation*}$

becomes the right-half complex plane and the class $\mathcal{S}_{q}^{\ast }$ reduces to the above-mentioned well-known class $\mathcal{S}^{\ast}$ of normalized starlike functions in $\mathbb{U}$ .

We note that the notation $\mathcal{S}_{q}^{\ast}$ was first used by Sahoo and Sharma (see ^[19]).

We now recall the familiar Mittag-Leffler function $E_{\alpha}(z)$ (see ^[14]) and its two-parameter extension $E_{\alpha, \beta}(z)$ having similar properties to those of the Mittag-Leffler function $E_{\alpha }(z)$ (see ^[28] and ^[29]), which are defined (as usual) by means of the following series:

$\begin{equation} E_{\alpha }(z) = \sum\limits_{n = 0}^{\infty }\frac{z^{n}}{\Gamma \left( \alpha n+1\right) } \ \ \ \ \ \left( z\in \mathbb{C} ;\;\alpha \gt 0\right) \end{equation}$

(1.8)

and

$\begin{equation} E_{\alpha ,\beta }(z) = \sum\limits_{n = 0}^{\infty }\frac{z^{n}}{\Gamma \left( \alpha n+\beta \right) } \ \ \ \ \ \left( z\in \mathbb{C} ;\; \alpha \gt 0;\; \beta \gt 0\right) , \end{equation}$

(1.9)

respectively. For a detailed account of the properties, generalizations and applications of the functions in (1.8) and (1.9), one may refer to ^[6,7,17,25].

The above-defined Mittag-Leffler functions $E_{\alpha }(z)$ and $E_{\alpha, \beta }(z)$ can be normalized as follows:

$\begin{equation*} {\bf E}_{\alpha ,\beta }\left( z\right) = z\Gamma\left( \beta \right) E_{\alpha ,\beta }(z) = \sum\limits_{n = 0}^{\infty }\frac{\Gamma \left( \beta \right) }{\Gamma\left( \alpha n+\beta \right) }z^{n+1}\qquad \left( z\in \mathbb{U};\; \alpha \gt 0;\;\beta \gt 0 \right) . \end{equation*}$

We note that

$\begin{equation} \left\{ \begin{array}{l} \left({\bf E}_{\alpha ,\beta }\right) _{0}\left( z\right) = z \\ \\ \left( {\bf E}_{\alpha ,\beta }\right) _{j}\left( z\right) = z+\sum\limits_{n = 1}^{j}\omega_{n}z^{n+1}\ \ \ \ (j\in \mathbb{N}), \end{array} \right. \end{equation}$

(1.10)

where

$\begin{equation*} \omega_{n} = \frac{\Gamma\left( \beta \right) }{\Gamma\left( \alpha n+\beta \right) }\ \ \ \ \ \ \ (\alpha \gt 0; \beta \gt 0 n\in \mathbb{N}). \end{equation*}$

Geometric properties including starlikeness, convexity and close-to-convexity for the Mittag-Leffler function $E_{\alpha, \beta}(z)$ were investigated by Bansal and Prajapat in ^[3] and, more recently, by Srivastava and Bansal (see ^[24]). In fact, the generalized Mittag-Leffler function $E_{\alpha, \beta}(z)$ and its extensions and generalizations continue to be used in many different contexts in geometric function theory (see, for details, ^[23]).

The $q$ -Mittag-Leffler function $\mathfrak{M}_{\alpha, \beta }\left(z; q\right)$ is normalized as follows (see, for example, ^[21]) :

$\begin{gather} \mathfrak{M}_{\alpha,\beta }\left( z;q\right) = z\Gamma _{q}\left( \beta \right) E_{\alpha ,\beta }(z) = \sum\limits_{n = 0}^{\infty }\frac{\Gamma _{q}\left( \beta \right) }{\Gamma _{q}\left( \alpha n+\beta \right)} z^{n+1}, \\ \left( z\in \mathbb{C} ;\;\alpha \gt 0;\;\beta \in \mathbb{C} \setminus \left\{ 0,-1,-2,...\right\} \right) . \end{gather}$

(1.11)

Some special cases of the normalized $q$ -Mittag-Leffler function $\mathfrak{M}_{\alpha, \beta }\left(z; q\right)$ are listed below:

$\begin{equation} \left\{ \begin{array}{lll} \mathfrak{M}_{0,\beta }\left( z;q\right) = \frac{z}{1-z} \\ \\ \mathfrak{M}_{1,1}\left( z;q\right) = ze_{q}^{z}\ \\ \\ \mathfrak{M}_{1,2}\left( z;q\right) = e_{q}^{z}-1 \\ \\ \mathfrak{M}_{1,3}\left( z;q\right) = \dfrac{\left( e_{q}^{z}-z-1\right) \left( 1+q\right) }{z} \\ \\ \mathfrak{M}_{1,4}\left( z;q\right) = \dfrac{\left( 1+q\right) \left( 1+q+q^{2}\right) }{z^{2}}\left( e_{q}^{z}-z-1-\frac{z^{2}}{1+q}\right) , \end{array} \right. \end{equation}$

(1.12)

where $e_{q}^{z}$ is one of the $q$ -analogues of the exponential function $e^{z},$ which is given by (see [25, p. 488, Eq. 6.3 (7)])

$\begin{equation} e_{q}^{z} = \sum\limits_{n = 0}^{\infty }\frac{z^{n}}{\Gamma _{q}\left( n+1\right) }. \end{equation}$

(1.13)

Recently, several results were given such as those related to partial sums of special functions, such as the Struve function ^[30], meromorphic functions (see ^[11] and ^[2]), the Bessel function ^[15], the Lommel function ^[4] and the Wright functions ^[5]. Several other works dealing with partial sums of various subclasses of the analytic function class $\mathcal{A}$ , the interested reader may refer (for example) to ^[12,16] and ^[26].

Motivated by the above-mentioned results, in this paper we investigate the ratio of the normalized $q$ -Mittag-Leffler function $\mathfrak{M}_{\alpha, \beta }\left(z; q\right)$ defined by $\left(1.11\right)$ to its sequence of partial sums:

$\begin{equation} \left\{ \begin{array}{l} \left( \mathfrak{M}_{\alpha ,\beta }\right) _{0}\left( z;q\right) = z \\ \\ \left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) = z+\sum\limits_{n = 1}^{j}K_{n}z^{n+1}\ \ \ \ (j\in \mathbb{N}), \end{array} \right. \end{equation}$

(1.14)

where

$\begin{equation*} K_{n} = \frac{\Gamma _{q}\left( \beta \right) }{\Gamma _{q}\left( \alpha n+\beta \right) }\ \ \ \ \ \ \ (\alpha \gt 0, \ \beta \gt 0 \ n\in \mathbb{N}). \end{equation*}$

We obtain the lower bounds on such ratios as those given below:

$\begin{align*} & \Re\left\{ \frac{\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) }{ \left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) }\right\} , \ \ \ \Re\left\{ \frac{\left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) }{\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) } \right\} \\ & \Re\left\{ \frac{D_{q}\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) }{D_{q}\left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) } \right\} , \ \ \ \ \Re\left\{ \frac{D_{q}\left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) }{D_{q}\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) }\right\} . \end{align*}$

2. Main results

The following lemma will be required in order to derive our main results.

Lemma 2.1. Let $q\in \left(0, 1\right), \ \alpha \geqq 1$ and $\beta \geqq 1$ . Then the function $\mathfrak{M}_{\alpha, \beta }\left(z; q\right)$ satisfies the following inequalities $:$

$\begin{equation} \left\vert \mathfrak{M}_{\alpha ,\beta }\left( z;q\right) \right\vert \leqq \frac{1+\left( q^{\beta }+q-3\right) q^{\beta +1}}{\left( 1-q^{\beta }\right) ^{2}q} \end{equation}$

(2.1)

and

$\begin{equation} \left\vert D_{q}\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) \right\vert \leqq \frac{6+q^{2\beta }+3q^{\beta +1}-5q^{\beta }+2q^{2}-7q}{\left( 1-q^{\beta }\right) ^{2}}. \end{equation}$

(2.2)

Proof. It is well-known that

$\begin{equation*} \Gamma _{q}\left( \alpha +\beta \right) \leqq \Gamma _{q}\left( \alpha n+\beta \right) . \end{equation*}$

Therefore, we have

$\begin{equation} \frac{\Gamma _{q}\left( \beta \right) }{\Gamma _{q}\left( \alpha n+\beta \right) }\leqq \frac{\Gamma _{q}\left( \beta \right) }{\Gamma _{q}\left( \alpha +\beta \right) } = \left( \frac{1}{\left[ \beta \right] }_{q}\right) _{n}. \end{equation}$

(2.3)

By making use of (1.3), (2.3) and well-known triangle inequality for $\left(z\in \mathbb{U}\right)$ , we find that

$\begin{align*} \left\vert \mathfrak{M}_{\alpha ,\beta }\left( z;q\right) \right\vert & = \left\vert z+\sum\limits_{n = 1}^{\infty }\frac{\Gamma _{q}\left( \beta \right) }{\Gamma _{q}\left( \alpha n+\beta \right) }z^{n+1}\right\vert \lt 1+\sum\limits_{n = 1}^{\infty }\frac{\Gamma _{q}\left( \beta \right) }{\Gamma _{q}\left( \alpha n+\beta \right) } \\ & \lt 1+\sum\limits_{n = 1}^{\infty }\left( \frac{1}{\left[ \beta \right] _{q} }\right) _{n} \\ & = 1+\frac{1}{\left[ \beta \right] _{q}}\sum\limits_{n = 1}^{\infty }\left( \frac{1}{\left[ \beta +1\right] _{q}}\right) _{n-1} \lt 1+\frac{1}{\left[ \beta \right] _{q}}\sum\limits_{n = 1}^{\infty }\left( \frac{1}{\left[ \beta+1 \right] _{q}}\right) ^{n-1} \\ & = 1+\frac{1}{\left[ \beta \right] _{q}}\sum\limits_{n = 0}^{\infty }\left( \frac{1}{\left[ \beta+1 \right] _{q}}\right) ^{n} = \frac{1+\left( q^{\beta }+q-3\right) q^{\beta +1}}{\left( 1-q^{\beta }\right) ^{2}q}. \end{align*}$

Hence, the inequality (2.1) is proved. Similarly, we can prove the inequality (2.2).

Let $w(z)$ denote an analytic function in $\mathbb{U}$ . In the proof of our main results, the following well-known result will be used frequently:

$\begin{equation*} \Re\left\{ \frac{1+w\left( z\right) }{1-w\left( z\right) }\right\} \gt 0 \end{equation*}$

if and only if

$\begin{equation*} \left\vert w\left( z\right) \right\vert \lt 1 \ \ \ \left( z\in \mathbb{U}\right) . \end{equation*}$

Theorem 2.2. Let $q\in \left(0, 1\right), \ \alpha \geqq 1$ and $\beta \geqq \frac{1+\sqrt{5}}{2}$ . Then

$\begin{equation} \Re\left\{ \frac{\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) }{\left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) }\right\} \geqq \frac{q^{\beta +1}\left( q^{\beta }-q-1\right) +2q-1}{\left( 1-q^{\beta }\right) ^{2}q}\mathit \ \ \ \left( z\in \mathbb{U}\right) \end{equation}$

(2.4)

and

$\begin{equation} \Re\left\{ \frac{\left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) }{\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) } \right\} \geqq \frac{\left( 1-q^{\beta }\right) ^{2}q}{1+q^{\beta +1}\left( q^{\beta }+q-3\right) }\mathit \ \ \ \left( z\in \mathbb{U}\right) . \end{equation}$

(2.5)

Proof. From the inequality (2.1), we obtain

$\begin{equation*} 1+\sum\limits_{n = 1}^{\infty }K_{n}\leqq \frac{1+\left( q^{\beta }+q-3\right) q^{\beta +1}}{\left( 1-q^{\beta }\right) ^{2}q}, \ \ \text{where }K_{n} = \frac{\Gamma _{q}\left( \beta \right) }{\Gamma _{q}\left( \alpha n+\beta \right) }\quad (n\in \mathbb{N}), \end{equation*}$

which is equivalent to

$\begin{equation*} \frac{\left( 1-q^{\beta }\right) ^{2}q}{1-q-q^{\beta +1}+q^{\beta +2}} \sum\limits_{n = 1}^{\infty }K_{n}\leqq 1. \end{equation*}$

In order to prove the inequality (2.4), we set

$\begin{align} & \frac{\left( 1-q^{\beta }\right) ^{2}q}{1-q-q^{\beta +1}+q^{\beta +2}} \left[ \frac{\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) }{\left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) }-\frac{q^{\beta +1}\left( q^{\beta }-q-1\right) +2q-1}{\left( 1-q^{\beta }\right) ^{2}q} \right] \\ &\qquad = \frac{1+\sum\limits_{n = 1}^{j}K_{n}z^{n}+\frac{\left( 1-q^{\beta }\right) ^{2}q}{1-q-q^{\beta +1}+q^{\beta +2}}\sum\limits_{n = j+1}^{\infty }K_{n}z^{n} }{1+\sum\limits_{n = 1}^{j}K_{n}z^{n}} \\ & \qquad = \frac{1+w\left( z\right) }{1-w\left( z\right) }, \end{align}$

(2.6)

where

$\begin{equation*} w\left( z\right) = \frac{\frac{\left( 1-q^{\beta }\right) ^{2}q}{1-q-q^{\beta +1}+q^{\beta +2}}\sum\limits_{n = j+1}^{\infty }K_{n}z^{n}}{ 2+2\sum\limits_{n = 1}^{j}K_{n}z^{n}+\frac{\left( 1-q^{\beta }\right) ^{2}q}{ 1-q-q^{\beta +1}+q^{\beta +2}}\sum\limits_{n = j+1}^{\infty }K_{n}z^{n}} \end{equation*}$

and

$\begin{equation*} \left\vert w\left( z\right) \right\vert \lt \frac{\frac{\left( 1-q^{\beta }\right) ^{2}q}{1-q-q^{\beta +1}+q^{\beta +2}}\sum\limits_{n = j+1}^{\infty }K_{n}}{2-2\sum\limits_{n = 1}^{j}K_{n}-\frac{\left( 1-q^{\beta }\right) ^{2}q }{1-q-q^{\beta +1}+q^{\beta +2}}\sum\limits_{n = j+1}^{\infty }K_{n}}. \end{equation*}$

The inequality $\left\vert w\left(z\right) \right\vert < 1$ holds true if and only if

$\begin{equation*} \frac{\left( 1-q^{\beta }\right) ^{2}q}{1-q-q^{\beta +1}+q^{\beta +2}} \sum\limits_{n = j+1}^{\infty }K_{n}\leqq 1-\sum\limits_{n = 1}^{j}K_{n} \end{equation*}$

or, equivalently,

$\begin{equation} \sum\limits_{n = 1}^{j}K_{n}+\frac{\left( 1-q^{\beta }\right) ^{2}q}{ 1-q-q^{\beta +1}+q^{\beta +2}}\sum\limits_{n = j+1}^{\infty }K_{n}\leqq 1. \end{equation}$

(2.7)

To prove (2.7), it suffices to show that its left-hand side is bounded above by

$\begin{equation*} \frac{\left( 1-q^{\beta }\right) ^{2}q}{1-q-q^{\beta +1}+q^{\beta +2}} \sum\limits_{n = 1}^{\infty }K_{n}, \end{equation*}$

which is equivalent to

$\begin{equation} \frac{q^{\beta +1}\left( q^{\beta }-q-1\right) +2q-1}{1-q-q^{\beta +1}+q^{\beta +2}}\sum\limits_{n = 1}^{j}K_{n}\geqq 0. \end{equation}$

(2.8)

We see that the inequality $\left(2.8\right)$ holds true for $\beta \geqq \frac{1+\sqrt{5}}{2}.$

We next use the same method to prove the inequality (2.5). Consider the function $w(z)$ given by

$\begin{align} & \frac{1+q^{\beta +1}\left( q^{\beta }+q-3\right) }{1-q-q^{\beta +1}+q^{\beta +2}}\left[ \frac{\left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) }{\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) }- \frac{\left( 1-q^{\beta }\right) ^{2}q}{1+q^{\beta +1}\left( q^{\beta }+q-3\right) }\right] \\ & \qquad = \frac{1+\sum\limits_{n = 1}^{j}K_{n}z^{n}-\frac{\left( 1-q^{\beta }\right) ^{2}q}{1-q-q^{\beta +1}+q^{\beta +2}}\sum\limits_{n = j+1}^{\infty }K_{n}z^{n} }{1+\sum\limits_{n = 1}^{\infty }K_{n}z^{n}} \\ &\qquad = \frac{1+w\left( z\right) }{1-w\left( z\right) }, \end{align}$

(2.9)

where

$\begin{equation*} w\left( z\right) = \frac{-\frac{1+q^{\beta +1}\left( q^{\beta }+q-3\right) }{ 1-q-q^{\beta +1}+q^{\beta +2}}\sum\limits_{n = j+1}^{\infty }K_{n}z^{n}}{ 2+2\sum\limits_{n = 1}^{j}K_{n}z^{n}-\frac{q^{2\beta +1}-q^{\beta +2}-q^{\beta +1}+2q-1}{1-q-q^{\beta +1}+q^{\beta +2}}\sum\limits_{n = j+1}^{\infty }K_{n}z^{n}} \end{equation*}$

and

$\begin{equation*} \left\vert w\left( z\right) \right\vert \lt \frac{\frac{1+q^{\beta +1}\left( q^{\beta }+q-3\right) }{1-q-q^{\beta +1}+q^{\beta +2}}\sum\limits_{n = j+1}^{ \infty }K_{n}}{2-2\sum\limits_{n = 1}^{j}K_{n}-\frac{q^{2\beta +1}-q^{\beta +2}-q^{\beta +1}+2q-1}{1-q-q^{\beta +1}+q^{\beta +2}}\sum\limits_{n = j+1}^{ \infty }K_{n}}. \end{equation*}$

Therefore, we get $\left\vert w\left(z\right) \right\vert < 1$ if and only if

$\begin{equation*} \frac{\left( 1-q^{\beta }\right) ^{2}q}{1-q-q^{\beta +1}+q^{\beta +2}} \sum\limits_{n = j+1}^{\infty }K_{n}+\sum\limits_{n = 1}^{j}K_{n}\leqq 1. \end{equation*}$

As the left-hand side of the last inequality is bounded above by

$\begin{equation*} \frac{\left( 1-q^{\beta }\right) ^{2}q}{1-q-q^{\beta +1}+q^{\beta +2}} \sum\limits_{n = 1}^{\infty }K_{n}, \end{equation*}$

we are led immediately to the assertion (2.5) of Theorem 2.2. Now we have completed the proof of Theorem 2.2.

In its special case, if we let $q\rightarrow 1-,$ Theorem 2.2 yields the following corollary.

Corollary 2.3. (see ^[18]) Let $\alpha \geqq 1$ and $\beta \geqq \frac{1+\sqrt{5}}{2}$ . Then

$\begin{equation*} \Re\left\{ \frac{E_{\alpha ,\beta }\left( z\right) }{\left( E_{\alpha ,\beta }\right) _{j}\left( z\right) }\right\} \geqq \frac{\beta ^{2}-\beta -1 }{\beta ^{2}}\mathit \ \ \ \ \left( z\in \mathbb{U}\right) \end{equation*}$

and

$\begin{equation*} \Re\left\{ \frac{\left( E_{\alpha ,\beta }\right) _{j}\left( z\right) }{E_{\alpha ,\beta }\left( z\right) }\right\} \geqq \frac{\beta ^{2}}{\beta ^{2}+\beta +1}\mathit \ \ \ \left( z\in \mathbb{U}\right) . \end{equation*}$

We next turn to the ratios involving derivatives.

Theorem 2.4. Let $q\in \left(0, 1\right),$ $\alpha \geqq 1$ and $\beta \geqq \frac{3+\sqrt{17}}{2}$ . Then

$\begin{equation} \Re\left\{ \frac{D_{q}\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) }{ D_{q}\left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) } \right\} \geqq \frac{q^{2\beta }-3q^{\beta +1}+q^{\beta }-2q^{2}+7q-4}{\left( 1-q^{\beta }\right) ^{2}}\mathit \ \ \ \left( z\in \mathbb{U}\right) \end{equation}$

(2.10)

and

$\begin{equation} \Re\left\{ \frac{D_{q}\left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) }{D_{q}\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) } \right\} \geqq \frac{\left( 1-q^{\beta }\right) ^{2}q}{6+q^{2\beta }+3q^{\beta +1}-5q^{\beta }+2q^{2}-7q}\mathit \ \ \ \left( z\in \mathbb{U} \right) . \end{equation}$

(2.11)

Proof. From the inequality (2.2), we have

$\begin{equation} 1+\sum\limits_{n = 1}^{\infty }\left[ n+1\right] _{q}K_{n}\leqq \frac{ 6+q^{2\beta }+3q^{\beta +1}-5q^{\beta }+2q^{2}-7q}{\left( 1-q^{\beta }\right) ^{2}}, \end{equation}$

(2.12)

where

$\begin{equation*} K_{n} = \frac{\Gamma _{q}\left( \beta \right) }{\Gamma _{q}\left( \alpha n+\beta \right) }\qquad (n\in \mathbb{N}). \end{equation*}$

Equivalently, we can rewrite the condition in $\left(2.12\right)$ as follows:

$\begin{equation*} \frac{\left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q} \sum\limits_{n = 1}^{\infty }\left[ n+1\right] _{q}K_{n}\leqq 1. \end{equation*}$

In order to prove the inequality (2.10), we consider the function $w\left(z\right)$ defined by

$\begin{align} & \frac{\left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q}\left[ \frac{D_{q}\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) }{D_{q}\left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) } \right. \\ &\qquad \qquad \left. -\frac{q^{2\beta }-3q^{\beta +1}+q^{\beta }-2q^{2}+7q-4}{\left( 1-q^{\beta }\right) ^{2}}\right] \\ &\qquad = \frac{1+\sum\limits_{n = 1}^{j}\left[ n+1\right] _{q}K_{n}z^{n}+\frac{ \left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q} \sum\limits_{n = j+1}^{\infty }\left[ n+1\right] _{q}K_{n}z^{n}}{ 1+\sum\limits_{n = 1}^{j}\left[ n+1\right] _{q}K_{n}z^{n}} \\ &\qquad = \frac{1+w\left( z\right) }{1-w\left( z\right) }. \end{align}$

(2.13)

From (2.13), we have

$\begin{equation*} w\left( z\right) = \frac{\frac{\left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q}\sum\limits_{n = j+1}^{\infty }\left[ n+1\right] _{q}K_{n}z^{n}}{2+2\sum\limits_{n = 1}^{j}\left[ n+1\right] _{q}K_{n}z^{n}+ \frac{\left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q} \sum\limits_{n = j+1}^{\infty }\left[ n+1\right] _{q}K_{n}z^{n}} \end{equation*}$

or, equivalently

$\begin{equation*} w\left( z\right) = \frac{\frac{\left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q}\sum\limits_{n = j+1}^{\infty }\left[ n+1\right] _{q}K_{n}}{2-2\sum\limits_{n = 1}^{j}\left[ n+1\right] _{q}K_{n}-\frac{\left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q} \sum\limits_{n = j+1}^{\infty }\left[ n+1\right] _{q}K_{n}}. \end{equation*}$

The inequality $\left\vert w\left(z\right) \right\vert < 1$ holds true if and only if

$\begin{equation*} \sum\limits_{n = 1}^{j}\left[ n+1\right] _{q}K_{n}+\frac{\left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q}\sum\limits_{n = j+1}^{ \infty }\left[ n+1\right] _{q}K_{n}\leqq 1. \end{equation*}$

The left-hand side of the above inequality is bounded above by

$\begin{equation*} \frac{\left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q} \sum\limits_{n = 1}^{\infty }\left[ n+1\right] _{q}K_{n}, \end{equation*}$

which is equivalent to

$\begin{equation} \frac{q^{2\beta }-3q^{\beta +1}+q^{\beta }-2q^{2}+7q-4}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q}\sum\limits_{n = 1}^{j}\left[ n+1\right] _{q}K_{n}\geqq 0. \end{equation}$

(2.14)

The inequality in $\left(2.14\right)$ holds true for $\beta \geqq \frac{3+\sqrt{17}}{2}.$

We next use the same method to prove the inequality (2.5). Consider the function $w(z)$ given by

$\begin{align} & \frac{6+q^{2\beta }+3q^{\beta +1}-5q^{\beta }+2q^{2}-7q}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q}\left[ \frac{D_{q}\left( \mathfrak{M}_{\alpha ,\beta }\right) _{j}\left( z;q\right) }{D_{q}\mathfrak{M}_{\alpha ,\beta }\left( z;q\right) }\right. \\ & \qquad \qquad \left. -\frac{\left( 1-q^{\beta }\right) ^{2}}{6+q^{2\beta }+3q^{\beta +1}-5q^{\beta }+2q^{2}-7q}\right] \\ &\qquad = \frac{1+\sum\limits_{n = 1}^{j}\left[ n+1\right] _{q}K_{n}z^{n}-\frac{ \left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q} \sum\limits_{n = j+1}^{\infty }\left[ n+1\right] _{q}K_{n}z^{n}}{ 1+\sum\limits_{n = 1}^{\infty }\left[ n+1\right] _{q}K_{n}z^{n}} \\ &\qquad = \frac{1+w\left( z\right) }{1-w\left( z\right) }. \end{align}$

(2.15)

By using Eq. (2.15), we obtain

$\begin{equation*} w\left( z\right) = \frac{-\frac{6+q^{2\beta }+3q^{\beta +1}-5q^{\beta }+2q^{2}-7q}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q}\sum\limits_{n = j+1}^{ \infty }\left[ n+1\right] _{q}K_{n}z^{n}}{2+2\sum\limits_{n = 1}^{j}\left[ n+1 \right] _{q}K_{n}z^{n}-\frac{q^{^{2\beta }}-3q^{^{\beta +1}}+q^{\beta }-2q^{2}+7q-4}{5+3q^{\beta }-3q^{\beta }+2q^{2}-7q}\sum\limits_{n = j+1}^{ \infty }\left[ n+1\right] _{q}K_{n}z^{n}}, \end{equation*}$

which is equivalent to

$\begin{equation*} \left\vert w\left( z\right) \right\vert \lt \frac{\frac{6+q^{2\beta }+3q^{\beta +1}-5q^{\beta }+2q^{2}-7q}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q} \sum\limits_{n = j+1}^{\infty }\left[ n+1\right] _{q}K_{n}}{ 2-2\sum\limits_{n = 1}^{j}\left[ n+1\right] _{q}K_{n}-\frac{q^{^{2\beta }}-3q^{^{\beta +1}}+q^{\beta }-2q^{2}+7q-4}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q}\sum\limits_{n = j+1}^{\infty }\left[ n+1\right] _{q}K_{n}} \end{equation*}$

The inequality $\left\vert w\left(z\right) \right\vert < 1$ holds true if and only if

$\begin{equation*} \frac{2\left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q }\sum\limits_{n = j+1}^{\infty }\left[ n+1\right] _{q}K_{n}\leqq 2-2\sum\limits_{n = 1}^{j}\left[ n+1\right] _{q}K_{n} \end{equation*}$

or, equivalently,

$\begin{equation} \sum\limits_{n = 1}^{j}\left[ n+1\right] _{q}K_{n}+\frac{\left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q}\sum\limits_{n = j+1}^{ \infty }\left[ n+1\right] _{q}K_{n}\leqq 1. \end{equation}$

(2.16)

It now suffices to show that the left-hand side of (2.16) is bounded above by

$\begin{equation*} \frac{\left( 1-q^{\beta }\right) ^{2}}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q} \sum\limits_{n = j+1}^{\infty }\left[ n+1\right] _{q}K_{n}, \end{equation*}$

which is equivalent to

$\begin{equation*} \frac{q^{^{2\beta }}-3q^{^{\beta +1}}+q^{\beta }-2q^{2}+7q-4}{5+3q^{\beta +1}-3q^{\beta }+2q^{2}-7q}\sum\limits_{n = j+1}^{\infty }\left[ n+1\right] _{q}K_{n}\geqq 0. \end{equation*}$

This last inequality holds true for $\beta \geqq \frac{3+\sqrt{17}}{2}.$ Hence we complete the proof of Theorem 2.4.

Upon letting $q\rightarrow 1-$ , Theorem 2.4 yields the following known result.

Corollary 2.5. (see ^[18]) Let $\alpha \geqq 1$ and $\beta \geqq \frac{1+\sqrt{5}}{2}$ . Then

$\begin{equation*} \Re\left\{ \frac{E_{\alpha ,\beta }^{\prime }\left( z\right) }{\left( E_{\alpha ,\beta }\right) _{j}^{\prime }\left( z\right) }\right\} \geqq \frac{ \beta ^{2}-3\beta -2}{\beta ^{2}}\mathit \ \ \ \left( z\in \mathbb{U}\right) \end{equation*}$

and

$\begin{equation*} \Re\left\{ \frac{\left( E_{\alpha ,\beta }\right) _{j}^{\prime }\left( z\right) }{E_{\alpha ,\beta }^{\prime }\left( z\right) }\right\} \geqq \frac{ \beta ^{2}}{\beta ^{2}+3\beta +2}\mathit \ \ \ \left( z\in \mathbb{U}\right) . \end{equation*}$

Conflicts of interest

The authors declare no conflicts of interest.

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AIMS Mathematics

Partial sums of generalized q-Mittag-Leffler functions

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AIMS Mathematics

Partial sums of generalized q-Mittag-Leffler functions

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