Geometric Attributes of Analytic Functions Generated by Mittag-Leffler Function

Abstract

This study examines the necessary requirements for some analytic function subclasses, especially those associated with the generalized Mittag-Leffler function, to be classified as univalent function subclasses that are determined by particular geometric constraints. The core methodology revolves around the application of the Hadamard (or convolution) product involving a normalized Mittag-Leffler function ${\mathbb{M}}_{\kappa ,\chi }\left(\zeta \right)$, leading to the definition of a new linear operator ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right).$ We investigate inclusion results in the recently defined subclasses $\tilde{\mathsf{\Xi }}(\varpi ,\varrho ),\widehat{\mathfrak{L}}(\varpi ,\varrho ),\widehat{\mathfrak{K}}(\varpi ,\varrho )$ and $\widehat{\mathfrak{F}}(\varpi ,\varrho )$, which generalize the classical classes of starlike, convex, and close-to-convex functions. This is achieved by utilizing recent developments in the theory of univalent functions. In addition, we examine the behavior of functions from the class ${\mathfrak{R}}_{\theta }(E,V)$ under the action of the convolution operator ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right)$, establishing sufficient criteria for the resulting images to lie within the subclasses of analytic function. Also, certain mapping properties related to these subclasses are analyzed. In addition, the geometric features of an integral operator connected to the Mittag-Leffler function are examined. A few particular cases of our main findings are also mentioned and examined and the paper ends with the conclusions regarding the obtained results.

1. Overview and Preliminary Information

Let ${\mathfrak{E}}_{\kappa }$ be the function defined by

$${\mathfrak{E}}_{\kappa }\left(\zeta \right)=\sum _{\iota =0}^{\infty }{\displaystyle \frac{{\zeta }^{\iota }}{\Gamma (\kappa \iota +1)}}(\zeta \in \mathbb{C},\kappa \in \mathbb{C},\mathrm{Re}\left(\kappa \right)>0),$$

that was introduced by Mittag-Leffler [1]. Wiman [2] defined a general function ${\mathfrak{E}}_{\kappa ,\chi }$ generalizing the Mittag-Leffler function, that is

$${\mathfrak{E}}_{\kappa ,\chi }\left(\zeta \right)=\sum _{\iota =0}^{\infty }{\displaystyle \frac{{\zeta }^{\iota }}{\Gamma (\kappa \iota +\chi )}}(\zeta \in \mathbb{C},\kappa ,\chi \in \mathbb{C},\mathrm{Re}\left(\kappa \right)>0,\mathrm{Re}\left(\chi \right)>0).$$

Note that

• ${\mathfrak{E}}_{0,1}\left(\zeta \right)={\displaystyle \frac{1}{1-\zeta }},{\mathfrak{E}}_{1,1}\left(\zeta \right)=e^{\zeta },{\mathfrak{E}}_{1,2}\left(\zeta \right)={\displaystyle \frac{e^{\zeta }-1}{\zeta }};$
• ${\mathfrak{E}}_{1,3}\left(\zeta \right)={\displaystyle \frac{2}{{\zeta }^2}}(e^{\zeta }-\zeta -1),{\mathfrak{E}}_{1,4}\left(\zeta \right)={\displaystyle \frac{1}{{\zeta }^3}}[6(e^{\zeta }-1-\zeta )-3{\zeta }^2]$.

The Mittag-Leffler function naturally occurs when fractional-order differential and integral equations are solved, particularly when random walks, fractional generalization of kinetic equations, super-diffusive transport, Lévy functions, and complex systems are studied. Mittag-Leffler functions generalize the exponential function and naturally arise in fractional calculus, anomalous diffusion, and related applied fields. Their analytic flexibility makes them particularly suitable for defining convolution and integral operators that preserve starlike, convex, and other geometric subclasses, while providing richer parameter control than operators based on other special functions. This motivates the present study and highlights its broader theoretical and applied relevance. Numerous attributes of the generalized Mittag-Leffler function and Mittag-Leffler function are seen, for instance, in Bansal and Prajapat [3], Frasin et al. [4] and others. The generalized (Mittag-Leffler) function ${\mathfrak{E}}_{\kappa ,\chi }$ does not fall under the family $\mathfrak{A}$, where $\mathfrak{A}$ is a family of functions in which members take the form

$$h\left(\zeta \right)=\zeta +\sum _{\iota =2}^{\infty }{c}_{\iota }{\zeta }^{\iota },$$

(1)

which are univalent in $\mathcal{P}=\{\zeta :\zeta \in \mathbb{C}$ and $\left|\zeta \right|<1\}.$ Let us also assume that $\mathcal{T}$ is a subclass of $\mathfrak{A}$, which includes the function of the kind that follows:

$$\hslash \left(\zeta \right)=\zeta -\sum _{ı=2}^{\infty }{c}_{\iota }{\zeta }^{\iota },(c_{\iota }\ge 0).$$

(2)

The convolution of $h\left(\zeta \right)$ and $g\left(\zeta \right)$ is obtained by [5,6]

$$(h\ast g)\left(\zeta \right)=\zeta +\sum _{\iota =2}^{\infty }{c}_{\iota }{d}_{\iota }{\zeta }^{\iota }=(g\ast h)\left(\zeta \right),(\zeta \in \mathcal{P}),$$

where $h\left(\zeta \right)\in \mathfrak{A},$ defined by (1), and $g\left(\zeta \right)\in \mathfrak{A}$ given by

$$g\left(\zeta \right)=\zeta +\sum _{\iota =2}^{\infty }{d}_{\iota }{\zeta }^{\iota }.$$

According to Lashin et al. [7], a new class was introduced:

Definition 1 

([7]). A function $h\left(\zeta \right)\in \Lambda ({\rm Y},\Psi ,\hslash ,\varpi ,\varrho )$ if

$$\mathit{Re}\left\{{\displaystyle \frac{(1-\varpi )(h\ast {\rm Y})\left(\zeta \right)+\varpi (h\ast \Psi )\left(\zeta \right)}{(h\ast \Theta )\left(\zeta \right)}}\right\}>\varrho (h\in \mathfrak{A},\varpi \in [0,1],\varrho \in [0,1)),$$

where ${\rm Y},\Psi$ and Θ are given by

$${\rm Y}\left(\zeta \right)=\zeta +\sum _{\iota =2}^{\infty }{\alpha }_{\iota }{\zeta }^{\iota },\Psi \left(\zeta \right)=\zeta +\sum _{\iota =2}^{\infty }{\nu }_{\iota }{\zeta }^{\iota }\mathit{and}\Theta \left(\zeta \right)=\zeta +\sum _{\iota =2}^{\infty }{\gamma }_{\iota }{\zeta }^{\iota },({\alpha }_{\iota },{\nu }_{\iota },{\gamma }_{\iota }\ge 0,\zeta \in \mathcal{P}).$$

Also,

$$\widehat{\Lambda }({\rm Y},\Psi ,\Theta ,\varpi ,\varrho )=\Lambda ({\rm Y},\Psi ,\Theta ,\varpi ,\varrho )\cap \mathcal{T}.$$

There are additionally more basic subclasses of the class mentioned above. Here are a few instances of these situations:

Assuming that ${\rm Y}\left(\zeta \right)={\displaystyle \frac{\zeta }{{(1-\zeta )}^2}},\Psi \left(\zeta \right)={\displaystyle \frac{\zeta (1+\zeta )}{{(1-\zeta )}^3}}$ and $\Theta \left(\zeta \right)=\zeta$ in Definition 1, we have the subclass results.

Definition 2 

([8]). A function $h\left(\zeta \right)\in \mathsf{\Xi }(\varpi ,\varrho )$ if

$$\mathit{Re}\left\{h^{\prime }\left(\zeta \right)+\varpi \zeta h^{″}\left(\zeta \right)\right\}>\varrho ,(\varrho \in [0,1),\varpi \in [0,1],h\in \mathfrak{A}),$$

and (see [7,9])

$$\tilde{\Xi }(\varpi ,\varrho )=\Xi (\varpi ,\varrho )\cap \mathcal{T}.$$

Lemma 1 

([9], with $p=1$). A function $\hslash \left(\zeta \right)\in \tilde{\Xi }(\varpi ,\varrho )$ if and only if

$$\sum _{\iota =2}^{\infty }\iota (1+\varpi (\iota -1))c_{\iota }\le 1-\varrho ,(\varrho \in [0,1),\varpi \in [0,1],c_{\iota }\ge 0).$$

(3)

If we put ${\rm Y}\left(\zeta \right)={\displaystyle \frac{\zeta }{{(1-\zeta )}^2}},\Psi \left(\zeta \right)={\displaystyle \frac{\zeta (1+\zeta )}{{(1-\zeta )}^3}}$ and $\Theta \left(\zeta \right)={\displaystyle \frac{\zeta }{1-\zeta }}$ in Definition 1, we derive the following subclass.

Definition 3 

([10]). A function $h\left(\zeta \right)\in \mathfrak{L}(\varpi ,\varrho )$ if

$$\mathit{Re}\left\{{\displaystyle \frac{\zeta h^{\prime }\left(\zeta \right)+\varpi {\zeta }^2{h}^{″}\left(\zeta \right)}{h\left(\zeta \right)}}\right\}>\varrho ,(\varrho \in [0,1),\varpi \in [0,1],h\in \mathfrak{A}),$$

and (see [7,11])

$$\widehat{\mathfrak{L}}(\varpi ,\varrho )=\mathfrak{L}(\varpi ,\varrho )\cap \mathcal{T}.$$

Lemma 2 

([11]). A function $\hslash \left(\zeta \right)\in \widehat{\mathfrak{L}}(\varpi ,\varrho )$ if and only if

$$\sum _{\iota =2}^{\infty }\left((\iota -1)(\varpi \iota +1)+1-\varrho \right)c_{\iota }\le 1-\varrho ,(\varrho \in [0,1),\varpi \in [0,1],c_{\iota }\ge 0).$$

(4)

If we put ${\rm Y}\left(\zeta \right)={\displaystyle \frac{\zeta }{1-\zeta }},\Psi \left(\zeta \right)={\displaystyle \frac{\zeta }{{(1-\zeta )}^2}}$ and $\Theta \left(\zeta \right)=\zeta$ in Definition 1, then the results that follow the subclass are obtained.

Definition 4 

([12]). A function $h\left(\zeta \right)\in \mathfrak{K}(\varpi ,\varrho )$ if

$$\mathit{Re}\left\{{\displaystyle \frac{(1-\varpi )h\left(\zeta \right)}{\zeta }}+\varpi h^{\prime }\left(\zeta \right)\right\}>\varrho ,(\varrho \in [0,1),\varpi \in [0,1],h\in \mathfrak{A}),$$

and (see [7] and ([13], with $h\left(z\right)={\displaystyle \frac{z}{1-z}}$))

$$\widehat{\mathfrak{K}}(\varpi ,\varrho )=\mathfrak{K}(\varpi ,\varrho )\cap \mathcal{T}.$$

Lemma 3 

([7]). A function $\hslash \left(\zeta \right)\in \widehat{\mathfrak{K}}(\varpi ,\varrho )$ if and only if

$$\sum _{\iota =2}^{\infty }\left(\varpi (\iota -1)+1\right)c_{\iota }\le 1-\varrho ,(\varrho \in [0,1),\varpi \in [0,1],c_{\iota }\ge 0).$$

(5)

If we put ${\rm Y}\left(\zeta \right)={\displaystyle \frac{\zeta }{1-\zeta }},\Psi \left(\zeta \right)={\displaystyle \frac{\zeta (1+\zeta )}{{(1-\zeta )}^3}}$ and $\Theta \left(\zeta \right)={\displaystyle \frac{\zeta }{{(1-\zeta )}^2}}$ in Definition 1, then we derive the following subclass.

Definition 5 

([14]). A function $h\left(\zeta \right)\in \mathfrak{F}(\varpi ,\varrho )$ if

$$\mathit{Re}\left\{(1-\varpi ){\displaystyle \frac{h\left(\zeta \right)}{\zeta h^{\prime }\left(\zeta \right)}}+\varpi \left(1+{\displaystyle \frac{\zeta h^{″}\left(\zeta \right)}{h^{\prime }\left(\zeta \right)}}\right)\right\}>\varrho ,(\varrho \in [0,1),\varpi \in [0,1],h\in \mathfrak{A}),$$

and

$$\widehat{\mathfrak{F}}(\varpi ,\varrho )=\mathfrak{F}(\varpi ,\varrho )\cap \mathcal{T}.$$

Lemma 4 

([14]). A function $\hslash \left(\zeta \right)\in \widehat{\mathfrak{F}}(\varpi ,\varrho )$ if and only if

$$\sum _{\iota =2}^{\infty }\left((\iota -1)\left(\varpi \right(\iota +1)-1)+\iota (1-\varrho )\right)c_{\iota }\le 1-\varrho ,\left({\displaystyle \frac{1}{3}}\le \varpi \le 1,0\le \varrho <1,c_{\iota }\ge 0\right).$$

(6)

Remark 1. 

(i) 

Taking $\varpi =0$ in Definition 3, we derive ${\mathcal{S}}^{*}\left(\varrho \right)$, which contains the starlike function of order $\varrho (0\le \varrho <1)$ and ${\widehat{\mathcal{S}}}^{*}\left(\varrho \right)={\mathcal{S}}^{\ast }\left(\varrho \right)\cap \mathcal{T}$ [15];

(ii) 

Taking $\varpi =\varrho =0$ in Definition 3, we derive ${\mathcal{S}}^{*}$ which contains the starlike function and ${\widehat{\mathcal{S}}}^{\ast }={\mathcal{S}}^{\ast }\cap \mathcal{T}$ [16];

(iii) 

Taking $\varpi =1$ in Definition 5, we derive $\mathcal{K}\left(\varrho \right)$, which contains the convex function of order $\varrho (0\le \varrho <1)$ and $\widehat{\mathcal{K}}\left(\varrho \right)=\mathcal{K}\left(\varrho \right)\cap \mathcal{T}$ [15];

(iv) 

Taking $\varpi =1$ and $\varrho =0$ in Definition 5, we derive $\mathcal{K}$, which contains the convex function and $\widehat{\mathcal{K}}=\mathcal{K}\cap \mathcal{T}$ [16];

(v) 

Taking $\varpi =0$ in Definition 2 and $\varpi =1$ in Definition 4, we derive $\mathcal{C}\left(\varrho \right)$, which contains the close-to-convex function of order $\varrho (0\le \varrho <1)$ and $\widehat{\mathcal{C}}\left(\varrho \right)=\mathcal{C}\left(\varrho \right)\cap \mathcal{T}$ [17].

Definition 6 

([18]). A function $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$ if it satisfies

$$\left|{\displaystyle \frac{h^{\prime }\left(\zeta \right)-1}{(E-V)\theta -V(h^{\prime }\left(\zeta \right)-1)}}\right|<1(\theta \in {\mathbb{C}}^{*};-1\le V<E\le 1;h\in \mathfrak{A}).$$

Note that

(1) 

${\mathfrak{R}}_1(\sigma ,-\sigma )=\mathfrak{R}\left(\sigma \right)=\left\{h\in \mathfrak{A}:\left|{\displaystyle \frac{h^{\prime }\left(\zeta \right)-1}{h^{\prime }\left(\zeta \right)+1}}\right|<\sigma (\zeta \in \mathcal{P};0<\sigma \le 1)\right\}$ (see [19]);

(2) 

${\mathfrak{R}}_{e^{-i\lambda }cos\lambda }(1-2\sigma ,-1)={\mathfrak{R}}_{\lambda }\left(\sigma \right)=\left\{h\in \mathfrak{A}:\mathrm{Re}\left(e^{i\lambda }(h^{\prime }\left(\lambda \right)-\sigma )\right)>0\right.$

$\left.(\zeta \in \mathcal{P};|\lambda |<{\displaystyle \frac{\pi }{2}};0\le \sigma <1)\right\}$ (see [20]).

Lemma 5 

([18]). If $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, then

$$|c_{\iota }|\le {\displaystyle \frac{(E-V)\left|\theta \right|}{\iota }}(\iota \ge 2;\theta \in {\mathbb{C}}^{*};-1\le V<E\le 1;h\in \mathfrak{A}).$$

One innovative and interesting field of study is the use of special functions in geometric function theory. Mathematics, physics, and engineering all make substantial use of it. Geometric function theory relies heavily on the applications of the Fox–Wright function [21], the Wright function [22] and the generalized hypergeometric function [20,23,24]. The use of distribution series, including Poisson, Pascal, Borel, and others, to investigate the properties of geometric functions has been the subject of extensive contemporary research. A Poisson distribution series was introduced by Porwal [25] in 2014, and the required and sufficient criteria for this series that pertain to the classes of analytic univalent functions have been determined (see also [26,27]). Porwal [28] presented the generalized distribution series and examined its geometrical characteristics in relation to univalent functions in 2018 (see also [29]). Ali et al. [30] assessed a few conditions for the Touchard polynomial to belong to the analytic function subclasses in 2023. In 2024, Murugusundaramoorthy and Bulboacä [31] demonstrated that certain subclasses of analytic functions related to the Mittag-Leffler function exist in subclasses of spiral-like univalent functions.

The following normalization of the Mittag-Leffler function was defined by Bansal and Prajapat [3]:

$$\begin{array}{cc}\hfill {\mathbb{M}}_{\kappa ,\chi }\left(\zeta \right)& =\zeta \Gamma \left(\chi \right){\mathfrak{E}}_{\kappa ,\chi }\left(\zeta \right)=\zeta +\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}{\zeta }^{\iota }\hfill \\ & (\zeta \in \mathbb{C},\kappa ,\chi \in \mathbb{C},\mathrm{Re}(\kappa )>0,\mathrm{Re}(\chi )>0).\hfill \end{array}$$

(7)

We note that [3]

(i) 

${\mathbb{M}}_{0,1}\left(\zeta \right)={\displaystyle \frac{\zeta }{1-\zeta }}$ is convex in $\mathcal{P}=\{\zeta :|\zeta |<1\};$

(ii) 

${\mathbb{M}}_{1,2}\left(\zeta \right)=e^{\zeta }-1,{\mathbb{M}}_{1,3}\left(\zeta \right)={\displaystyle \frac{2}{\zeta }}(e^{\zeta }-\zeta -1),{\mathbb{M}}_{1,4}\left(\zeta \right)={\displaystyle \frac{1}{{\zeta }^2}}[6(e^{\zeta }-1-\zeta )-3{\zeta }^2]$ are starlike and univalent in ${\mathcal{P}}_{1/2}=\{\zeta :|\zeta |<1/2\};$

(iii) 

${\mathbb{M}}_{1,4}\left(\zeta \right)={\displaystyle \frac{1}{{\zeta }^2}}[6(e^{\zeta }-1-\zeta )-3{\zeta }^2]$ is convex in ${\mathcal{P}}_{1/2}$;

(iv) 

${\mathbb{M}}_{1,4}\left(\zeta \right)={\displaystyle \frac{1}{{\zeta }^2}}[6(e^{\zeta }-1-\zeta )-3{\zeta }^2]$ is starlike in $\mathcal{P}$.

Now we define the following new linear operators based on convolution. For real parameters $\kappa$ and $\chi$ with $\kappa ,\chi \notin \{0,-1,-2,\dots \}$ and ${\mathbb{M}}_{\kappa ,\chi }\left(\zeta \right)$ be given by (7), we define ${\mathbb{H}}_{\chi }^{\kappa }:\mathfrak{A}\to \mathfrak{A}$ as follows:

$${\mathbb{H}}_{\chi }^{\kappa }h\left(\zeta \right)=h\left(\zeta \right)\ast {\mathbb{M}}_{\kappa ,\chi }\left(\zeta \right)=\zeta +\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}{c}_{\iota }{\zeta }^{\iota }(\zeta \in \mathbb{C}).$$

(8)

Also, we introduce ${\aleph }_{\chi }^{\kappa }\left(\zeta \right)$ as follows:

$${\aleph }_{\chi }^{\kappa }\left(\zeta \right)=2\zeta -{\mathbb{M}}_{\kappa ,\chi }\left(\zeta \right)=\zeta -\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}{\zeta }^{\iota }.$$

(9)

Next, we define ${\mathfrak{I}}_{\chi }^{\kappa }:\mathcal{T}\to \mathcal{T}$ as follows:

$${\mathfrak{I}}_{\chi }^{\kappa }\hslash \left(\zeta \right)=\hslash \left(\zeta \right)\ast {\aleph }_{\chi }^{\kappa }\left(\zeta \right)=\zeta -\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}{c}_{\iota }{\zeta }^{\iota }(\zeta \in \mathbb{C};c_{\iota }\ge 0).$$

(10)

Additionally, we give the functions

$$\begin{array}{ccc}\hfill {\mathfrak{W}}_{\chi }^{\kappa }(\vartheta ;\zeta )& =& (1-\vartheta ){\mathbb{M}}_{\kappa ,\chi }\left(\zeta \right)+\vartheta \zeta {\left({\mathbb{M}}_{\kappa ,\chi }\left(\zeta \right)\right)}^{\prime }\hfill \\ & =& \zeta +\sum _{\iota =2}^{\infty }[1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}{\zeta }^{\iota }(\vartheta \ge 0),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right)& =& h\left(\zeta \right)\ast {\mathfrak{W}}_{\chi }^{\kappa }(\vartheta ;\zeta )\hfill \\ & =& \zeta +\sum _{\iota =2}^{\infty }[1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}{c}_{\iota }{\zeta }^{\iota }(\vartheta \ge 0).\hfill \end{array}$$

(11)

Additionally, we give the functions

$$\begin{array}{ccc}\hfill {\widehat{\mathfrak{W}}}_{\chi }^{\kappa }(\vartheta ;\zeta )& =& 2\zeta -{\mathfrak{M}}_{\chi }^{\kappa }(\vartheta ;\zeta )\hfill \\ & =& \zeta -\sum _{\iota =2}^{\infty }[1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}{\zeta }^{\iota }(\vartheta \ge 0).\hfill \end{array}$$

(12)

Also, we define ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }:\mathcal{T}\to \mathcal{T}$ by the convolution as

$$\begin{array}{ccc}\hfill {\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right)& =& {\widehat{\mathfrak{W}}}_{\chi }^{\kappa }(\vartheta ;\zeta )\ast \hslash \left(\zeta \right)\hfill \\ & =& \zeta -\sum _{\iota =2}^{\infty }[1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}{c}_{\iota }{\zeta }^{\iota },(c_{\iota }\ge 0;\vartheta \ge 0).\hfill \end{array}$$

(13)

In the present work, some requirements that a Mittag-Leffler function must have to be part of analytic function subclasses are discussed. Furthermore, we derive various inclusion relations between ${\mathfrak{R}}_{\theta }(E,V)$ and subclasses of analytic functions. Additionally, certain mapping properties related to these subclasses are analyzed. We also present the geometric properties of an integral operator assosiated with the Mittag-Leffler function. We also discuss and analyze some specific examples of our key conclusions.

In this paper, as long as otherwise specified, we are assuming that $\kappa ,\chi >0,$$\vartheta \ge 0,0\le \varpi \le 1,0\le \varrho <1,\theta \in {\mathbb{C}}^{*},-1\le V<E\le 1,\zeta \in \mathcal{P}.$

2. Inclusion Results

As indicated throughout our study, we will utilize the notation (7) to demonstrate our primary findings; therefore,

$$\begin{array}{ccc}\hfill {\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1& =& \sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}},\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill {\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1& =& \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)& =& \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}},\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)& =& \sum _{\iota =3}^{\infty }{\displaystyle \frac{\iota (\iota -1)(\iota -2)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill {\mathbb{M}}_{\kappa ,\chi }^{\left(4\right)}\left(1\right)& =& \sum _{\iota =4}^{\infty }{\displaystyle \frac{\iota (\iota -1)(\iota -2)(\iota -3)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill {\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds& =& \sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}.\hfill \end{array}$$

(19)

Theorem 1. 

The function ${\widehat{\mathfrak{W}}}_{\chi }^{\kappa }(\vartheta ;\zeta )\in \tilde{\Xi }(\varpi ,\varrho )$ if and only if

$$\vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[(1+\varpi )\vartheta +\varpi ]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1\le 1-\varrho .$$

(20)

Proof. 

Since ${\widehat{\mathfrak{W}}}_{\chi }^{\kappa }(\vartheta ;\zeta )$ is defined by (12), Lemma 1 states that we must demonstrate that

$$\sum _{\iota =2}^{\infty }\left[\iota (1+\varpi (\iota -1))\right][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\le 1-\varrho .$$

(21)

Given that the inequality (21)’s left side might be expressed as

$$\begin{array}{ccc}\hfill {\Omega }_1(\varpi ,\vartheta ;\kappa ,\chi )& =& \sum _{\iota =2}^{\infty }\left[\iota (1+\varpi (\iota -1))\right][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }[(1-\varpi )\iota +\vartheta (1-\varpi )\iota (\iota -1)+\varpi {\iota }^2+\vartheta \varpi {\iota }^2(\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \vartheta \varpi \sum _{\iota =3}^{\infty }{\displaystyle \frac{\iota (\iota -1)(\iota -2)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+[(1+\varpi )\vartheta +\varpi ]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & +& \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}.\hfill \end{array}$$

thus, with (15)–(17), we obtain

$$\begin{array}{ccc}\hfill {\Omega }_1(\varpi ,\vartheta ;\kappa ,\chi )& =& \vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[(1+\varpi )\vartheta +\varpi ]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1.\hfill \end{array}$$

Consequently, it may be determined that (21) holds. □

Applying Theorem 1 and $\vartheta =0$, we get the next outcome:

Corollary 1. 

The function function ${\aleph }_{\chi }^{\kappa }\left(\zeta \right)\in \tilde{\Xi }(\varpi ,\varrho )$ if and only if

$$\varpi {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1\le 1-\varrho .$$

(22)

Applying Theorem 1 and $\varpi =0$, we get the next outcome:

Corollary 2. 

The function ${\widehat{\mathfrak{W}}}_{\chi }^{\kappa }(\vartheta ;\zeta )\in \widehat{\mathcal{C}}\left(\varrho \right)$ if and only if

$$\vartheta {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1\le 1-\varrho .$$

(23)

Applying Corollary 2 and $\vartheta =0$, we get the next outcome:

Example 1. 

The function ${\aleph }_{\chi }^{\kappa }\left(\zeta \right)\in \widehat{\mathcal{C}}\left(\varrho \right)$ if and only if

$${\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1\le 1-\varrho .$$

(24)

Theorem 2. 

The function ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right),$ defined by (13), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\tilde{\Xi }(\varpi ,\varrho )$ if and only if

$$\vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{\left(4\right)}\left(1\right)+[(1+4\varpi )\vartheta +\varpi ]{\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[2(1+\varpi )\vartheta +2\varpi +1]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1\le 1-\varrho .$$

Proof. 

According to Lemma 1 and (13), we must demonstrate that

$$\sum _{\iota =2}^{\infty }\left[\iota (1+\varpi (\iota -1))\right][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\le 1-\varrho .$$

(25)

Since $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$, then $|c_{\iota }|\le \iota .$ The inequality (25) can be represented as

$$\begin{array}{ccc}& & \sum _{\iota =2}^{\infty }\left[\iota (1+\varpi (\iota -1))\right][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\hfill \\ & \le & \sum _{\iota =2}^{\infty }\iota \left[\iota (1+\varpi (\iota -1))\right][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }\iota [(1-\varpi )\iota +\vartheta (1-\varpi )\iota (\iota -1)+\varpi {\iota }^2+\vartheta \varpi {\iota }^2(\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }\left[\vartheta \varpi \iota (\iota -1)(\iota -2)(\iota -3)+\left[\right(1+4\varpi )\vartheta +\varpi ]\iota (\iota -1)(\iota -2)\right.\hfill \\ & +& \left.\left[2\right(1+\varpi )\vartheta +2\varpi +1]\iota (\iota -1)+\iota \right]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \vartheta \varpi \sum _{\iota =4}^{\infty }{\displaystyle \frac{\iota (\iota -1)(\iota -2)(\iota -3)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+[(1+4\varpi )\vartheta +\varpi ]\sum _{\iota =3}^{\infty }{\displaystyle \frac{\iota (\iota -1)(\iota -2)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & +& [2(1+\varpi )\vartheta +2\varpi +1]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}.\hfill \end{array}$$

Thus, with (15)–(18), we obtain

$$\begin{array}{ccc}\hfill \vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{\left(4\right)}\left(1\right)& +& [(1+4\varpi )\vartheta +\varpi ]{\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[2(1+\varpi )\vartheta +2\varpi +1]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)\hfill \\ & +& [{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]\le 1-\varrho .\hfill \end{array}$$

As a result, Theorem 2 is proved. □

Applying Theorem 2 and $\vartheta =0$, we get the next outcome:

Corollary 3. 

The function ${\mathfrak{I}}_{\chi }^{\kappa }\hslash \left(\zeta \right),$ defined by (10), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\tilde{\Xi }(\varpi ,\varrho )$ if and only if

$$\varpi {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[2\varpi +1]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1\le 1-\varrho .$$

Applying Theorem 2 and $\varpi =0$, we get the next outcome:

Corollary 4. 

The function ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right),$ given by (13), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\widehat{\mathcal{C}}\left(\varrho \right)$ if and only if

$$\vartheta {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[2\vartheta +1]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1\le 1-\varrho .$$

Applying Corollary 4 and $\vartheta =0$, we get the next outcome:

Example 2. 

The function ${\mathfrak{I}}_{\chi }^{\kappa }\hslash \left(\zeta \right),$ given by (10), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\widehat{\mathcal{C}}\left(\varrho \right)$ if and only if

$${\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1\le 1-\varrho .$$

Theorem 3. 

The function ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right),$ given by (13), maps $\hslash \left(\zeta \right)\in \widehat{\mathcal{K}}$ to $\tilde{\Xi }(\varpi ,\varrho )$ if and only if (20) satisfies.

Proof. 

According to Lemma 1 and (13), we must demonstrate that

$$\sum _{\iota =2}^{\infty }\left[\iota (1+\varpi (\iota -1))\right][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\le 1-\varrho .$$

Since $\hslash \left(\zeta \right)\in \widehat{\mathcal{K}}$, then $|c_{\iota }|\le 1.$ Since Theorem 3’s proof has similarities to that of Theorem 1, we omit it. □

Theorem 4. 

The function ${\widehat{\mathfrak{W}}}_{\chi }^{\kappa }(\vartheta ;\zeta )\in \widehat{\mathfrak{L}}(\varpi ,\varrho )$ if and only if

$$\vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[\varpi +\vartheta (1+\varpi )]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[1-\varrho \vartheta ]{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+\varrho (\vartheta -1){\mathbb{M}}_{\kappa ,\chi }\left(1\right)\le 2(1-\varrho ).$$

(26)

Proof. 

Given Lemma 2 and (12) of ${\widehat{\mathfrak{W}}}_{\chi }^{\kappa }(\vartheta ;\zeta )$, it is straightforward to demonstrate that

$$\sum _{\iota =2}^{\infty }[(\iota -1)(\varpi \iota +1)+1-\varrho ][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\le 1-\varrho .$$

(27)

The left-hand side of (27) might be represented as

$$\begin{array}{ccc}\hfill {\Omega }_2(\varpi ,\varrho ,\vartheta ,\kappa ,\chi )& =& \sum _{\iota =2}^{\infty }[(\iota -1)(\varpi \iota +1)+1-\varrho ][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }\left[\varpi \vartheta \iota {(\iota -1)}^2+\vartheta {(\iota -1)}^2+\varpi \iota (\iota -1)+[(1-\varrho )\vartheta +1](\iota -1)\right.\hfill \\ & +& \left.(1-\varrho )\right]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \vartheta \varpi \sum _{\iota =3}^{\infty }{\displaystyle \frac{\iota (\iota -1)(\iota -2)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+[\varpi +\vartheta (1+\varpi )]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & +& (1-\varrho \vartheta )\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+\varrho (\vartheta -1)\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}.\hfill \end{array}$$

Thus, with (14)–(17), we obtain

$$\begin{array}{ccc}\hfill {\Omega }_2(\varpi ,\varrho ,\vartheta ,\kappa ,\chi )& =& \vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[\varpi +\vartheta (1+\varpi )]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[1-\varrho \vartheta ][{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]\hfill \\ & +& \varrho (\vartheta -1)[{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1].\hfill \end{array}$$

Thus, it may be concluded that (27) is true. □

Applying Theorem 4 and $\vartheta =0$, we get the next outcome:

Corollary 5. 

The function ${\aleph }_{\chi }^{\kappa }\left(\zeta \right)\in \widehat{\mathfrak{L}}(\varpi ,\varrho )$ if and only if

$$\varpi {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-\varrho {\mathbb{M}}_{\kappa ,\chi }\left(1\right)\le 2(1-\varrho ).$$

(28)

Applying Theorem 4 and $\varpi =0$, we get the next outcome:

Corollary 6. 

The function ${\widehat{\mathfrak{W}}}_{\chi }^{\kappa }(\vartheta ;\zeta )\in {\widehat{\mathcal{S}}}^{*}\left(\varrho \right)$ if and only if

$$\vartheta {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[1-\varrho \vartheta ]{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+\varrho (\vartheta -1){\mathbb{M}}_{\kappa ,\chi }\left(1\right)\le 2(1-\varrho ).$$

Applying Corollary 6 and $\vartheta =\varrho =0$, we get the next outcome:

Example 3. 

The function ${\aleph }_{\chi }^{\kappa }\left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ if and only if

$${\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)\le 2.$$

Theorem 5. 

The function ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right)$ given by (13), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\widehat{\mathfrak{L}}(\varpi ,\varrho )$ if and only if

$$\vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{\left(4\right)}\left(1\right)+[\varpi +\vartheta (1+4\varpi )]{\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[1-\varrho \vartheta +2\varpi +2\vartheta (\varpi +1)]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+(1-\varrho ){\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)\le 2(1-\varrho ).$$

Proof. 

Given Lemma 2 and (13), it is straightforward to demonstrate that

$$\sum _{\iota =2}^{\infty }[(\iota -1)(\varpi \iota +1)+1-\varrho ][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\le 1-\varrho .$$

(29)

Since $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$, then $|c_{\iota }|\le \iota$. The inequality (29) can be represented as

$$\begin{array}{ccc}& & \sum _{\iota =2}^{\infty }[(\iota -1)(\varpi \iota +1)+1-\varrho ][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\hfill \\ & \le & \sum _{\iota =2}^{\infty }\iota [(\iota -1)(\varpi \iota +1)+1-\varrho ][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }\iota \left[\varpi \vartheta \iota {(\iota -1)}^2+\vartheta {(\iota -1)}^2+\varpi \iota (\iota -1)+[(1-\varrho )\vartheta +1](\iota -1)+(1-\varrho )\right]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \vartheta \varpi \sum _{\iota =4}^{\infty }{\displaystyle \frac{\iota (\iota -1)(\iota -2)(\iota -3)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+[\varpi +\vartheta (1+4\varpi )]\sum _{\iota =3}^{\infty }{\displaystyle \frac{\iota (\iota -1)(\iota -2)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & +& [1-\varrho \vartheta +2\varpi +2\vartheta (\varpi +1)]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+(1-\varrho )\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}.\hfill \end{array}$$

Thus, with (15)–(18), we obtain

$$\begin{array}{ccc}\hfill \vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{\left(4\right)}\left(1\right)& +& [\varpi +\vartheta (1+4\varpi )]{\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)\hfill \\ & +& [1-\varrho \vartheta +2\varpi +2\vartheta (\varpi +1)]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+(1-\varrho )[{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]\le 1-\varrho .\hfill \end{array}$$

As a result, Theorem 5 is proved. □

Applying Theorem 5 and $\vartheta =0$, we get the next outcome:

Corollary 7. 

The function ${\mathfrak{I}}_{\chi }^{\kappa }\hslash \left(\zeta \right),$ given by (10), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\widehat{\mathfrak{L}}(\varpi ,\varrho )$ if and only if

$$\varpi {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[1+2\varpi ]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+(1-\varrho ){\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)\le 2(1-\varrho ).$$

Applying Theorem 5 and $\varpi =0$, we get the next outcome:

Corollary 8. 

The function ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right)$ given by (13), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to ${\widehat{\mathcal{S}}}^{*}\left(\varrho \right)$ if and only if

$$\vartheta {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[1+\vartheta (2-\varrho )]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+(1-\varrho ){\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)\le 2(1-\varrho ).$$

Taking $\vartheta =\varrho =0$ in Corollary 8, we obtain the following result:

Example 4. 

The function ${\mathfrak{I}}_{\chi }^{\kappa }\hslash \left(\zeta \right),$ given by (10), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to ${\widehat{\mathcal{S}}}^{*}$ if and only if

$${\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)\le 2.$$

Theorem 6. 

The function ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right),$ given by (13), maps $\hslash \left(\zeta \right)\in \widehat{\mathcal{K}}$ to $\widehat{\mathfrak{L}}(\varpi ,\varrho )$ if and only if (26) is satisfied.

Proof. 

According to Lemma 2 and (13), we must demonstrate that

$$\sum _{\iota =2}^{\infty }[(\iota -1)(\varpi \iota +1)+1-\varrho ][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\le 1-\varrho .$$

Since $\hslash \left(\zeta \right)\in \widehat{\mathcal{K}}$, then $|c_{\iota }|\le 1.$ Since Theorem 6’s proof has similarities to that of Theorem 4, we omit it. □

Theorem 7. 

The function ${\widehat{\mathfrak{W}}}_{\chi }^{\kappa }(\vartheta ;\zeta )\in \widehat{\mathfrak{K}}(\varpi ,\varrho )$ if and only if

$$\vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[\varpi +\vartheta (1-\varpi )]{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+[1+(\varpi -1)\vartheta -\varpi ]{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1\le 1-\varrho .$$

(30)

Proof. 

It is easy to show that, given Lemma 3 and (12),

$$\sum _{\iota =2}^{\infty }[(\iota -1)\varpi +1][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\le 1-\varrho .$$

(31)

Inequality (31)’s left-hand side might be represented as

$$\begin{array}{ccc}\hfill {\Omega }_3(\varpi ,\vartheta ,\kappa ,\chi )& =& \sum _{\iota =2}^{\infty }[\varpi (\iota -1)+1][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }[\varpi \vartheta \iota (\iota -1)+[\varpi +\vartheta (1-\varpi )]\iota +(1-\vartheta +\varpi (\vartheta -1))]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \varpi \vartheta \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+[\varpi +\vartheta (1-\varpi )]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & +& [1-\vartheta +\varpi (\vartheta -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}.\hfill \end{array}$$

Using (14)–(16), we thus get

$$\begin{array}{ccc}\hfill {\Omega }_3(\varpi ,\vartheta ,\kappa ,\chi )& =& \varpi \vartheta {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[\varpi +\vartheta (1-\varpi )][{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]+[1-\vartheta +\varpi (\vartheta -1)][{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1].\hfill \end{array}$$

Therefore, (31) can be considered true. □

Applying Theorem 7 and $\vartheta =0$, we get the next outcome:

Corollary 9. 

The function ${\aleph }_{\chi }^{\kappa }\left(\zeta \right)\in \widehat{\mathfrak{K}}(\varpi ,\varrho )$ if and only if

$$\varpi {\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+(1-\varpi ){\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1\le 1-\varrho .$$

(32)

Applying Theorem 7 and $\varpi =1$, we get the next outcome:

Corollary 10. 

The function ${\widehat{\mathfrak{W}}}_{\chi }^{\kappa }(\vartheta ;\zeta )\in \widehat{\mathcal{C}}\left(\varrho \right)$ if and only if (23) is satisfied.

Taking $\vartheta =0$ in Corollary 10, we obtain the following result:

Example 5. 

The function ${\aleph }_{\chi }^{\kappa }\left(\zeta \right)\in \widehat{\mathcal{C}}\left(\varrho \right)$ if and only if (24) is satisfied.

Theorem 8. 

The function ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right)$ defined by (13), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\widehat{\mathfrak{K}}(\varpi ,\varrho )$ if and only if (20) is satisfied.

Proof. 

It is easy to show that, given Lemma 3 and (13),

$$\sum _{\iota =2}^{\infty }[(\iota -1)\varpi +1][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\le 1-\varrho .$$

Since $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$, then $|c_{\iota }|\le \iota$. The proof of Theorem 8 is omitted because it is comparable to that of Theorem 1. □

Applying Theorem 8 and $\vartheta =0$, we get the next outcome:

Corollary 11. 

The function ${\mathfrak{I}}_{\chi }^{\kappa }\hslash \left(\zeta \right),$ defined by (10), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\widehat{\mathfrak{K}}(\varpi ,\varrho )$ if and only if (22) is satisfied.

Applying Theorem 8 and $\varpi =1$, we get the next outcome:

Corollary 12. 

The function ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right)$ defined by (13), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\widehat{\mathcal{C}}\left(\varrho \right)$ if and only if

$$\vartheta {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[2\vartheta +1]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1\le 1-\varrho .$$

Taking $\vartheta =0$ in Corollary 12, we obtain the following result:

Example 6. 

The function ${\mathfrak{I}}_{\chi }^{\kappa }\hslash \left(\zeta \right),$ defined by (10), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\widehat{\mathcal{C}}\left(\varrho \right)$ if and only if

$${\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1\le 1-\varrho .$$

Theorem 9. 

The function ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right),$ defined by (13), maps $\hslash \left(\zeta \right)\in \widehat{\mathcal{K}}$ to $\widehat{\mathfrak{K}}(\varpi ,\varrho )$ if and only if (30) is satisfied.

Proof. 

According to Lemma 3 and (13), we must demonstrate that

$$\sum _{\iota =2}^{\infty }[(\iota -1)\varpi +1][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\le 1-\varrho .$$

Since $\hslash \left(\zeta \right)\in \widehat{\mathcal{K}}$, then $|c_{\iota }|\le 1.$ Since Theorem 9’s proof has similarities to that of Theorem 7, we omit it. □

Theorem 10. 

The function ${\widehat{\mathfrak{W}}}_{\chi }^{\kappa }(\vartheta ;\zeta )\in \widehat{\mathfrak{F}}(\varpi ,\varrho )$ if and only if

$$\begin{array}{cc}\hfill \vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+& [\varpi +\vartheta (2\varpi -\varrho )]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[1-\varrho +(1-\varpi )(\vartheta -1)]{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)\hfill \\ \hfill & -(1-\varpi )(\vartheta -1){\mathbb{M}}_{\kappa ,\chi }\left(1\right)\le 2(1-\varrho ).\hfill \end{array}$$

(33)

Proof. 

It is easy to show that, given Lemma 4 and (12),

$$\sum _{\iota =2}^{\infty }[(\iota -1)(\varpi (\iota +1)-1)+\iota (1-\varrho )][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\le 1-\varrho .$$

(34)

Inequality (34)’s left-hand side might be represented as

$$\begin{array}{ccc}\hfill {\Omega }_4(\varpi ,\varrho ,\vartheta ,\kappa ,\chi )& =& \sum _{\iota =2}^{\infty }[(\iota -1)(\varpi (\iota +1)-1)+\iota (1-\varrho )][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }[\varpi \vartheta \iota (\iota -1)(\iota -2)+[\varpi +\vartheta (2\varpi -\varrho )]\iota (\iota -1)+[(1-\varpi )(\vartheta -1)+1-\varrho ]\iota \hfill \\ & -& (1-\varpi )(\vartheta -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \varpi \vartheta \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)(\iota -2)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+[\varpi +\vartheta (2\varpi -\varrho )]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & +& [(1-\varpi )(\vartheta -1)+1-\varrho ]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & -& (1-\varpi )(\vartheta -1)\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}.\hfill \end{array}$$

Using (14)–(17), we thus get

$$\begin{array}{ccc}\hfill {\Omega }_4(\varpi ,\varrho ,\vartheta ,\kappa ,\chi )& =& \varpi \vartheta {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[\varpi +\vartheta (2\varpi -\varrho )]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)\hfill \\ & +& [(1-\varpi )(\vartheta -1)+1-\varrho ][{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]-(1-\varpi )(\vartheta -1)[{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1].\hfill \end{array}$$

Therefore, (34) can be considered true. □

Applying Theorem 10 and $\vartheta =0$, we get the next outcome:

Corollary 13. 

The function ${\aleph }_{\chi }^{\kappa }\left(\zeta \right)\in \widehat{\mathfrak{F}}(\varpi ,\varrho )$ if and only if

$$\begin{array}{c}\hfill \varpi {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[\varpi -\varrho ]{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+(1-\varpi ){\mathbb{M}}_{\kappa ,\chi }\left(1\right)\le 2(1-\varrho ).\end{array}$$

(35)

Applying Theorem 10 and $\varpi =1$, we get the next outcome:

Corollary 14. 

The function ${\widehat{\mathfrak{W}}}_{\chi }^{\kappa }(\vartheta ;\zeta )\in \widehat{\mathcal{K}}\left(\varrho \right)$ if and only if

$$\vartheta {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[1+\vartheta (2-\varrho )]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[1-\varrho ]{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)\le 2(1-\varrho ).$$

(36)

Taking $\vartheta =\varrho =0$ in Corollary 14, we get the next result:

Example 7. 

The function ${\aleph }_{\chi }^{\kappa }\left(\zeta \right)\in \widehat{\mathcal{K}}$ if and only if

$${\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)\le 2.$$

(37)

Theorem 11. 

The function ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right),$ defined by (13), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\widehat{\mathfrak{F}}(\varpi ,\varrho )$ if and only if

$$\begin{array}{cc}\hfill \vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{\left(4\right)}\left(1\right)& +[\varpi (1+5\vartheta )-\vartheta \varrho ]{\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)\hfill \\ \hfill & +[(1+2\vartheta )(2\varpi -\varrho )+(1-\varpi )(\vartheta -1)+1]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+(1-\varrho ){\mathbb{M}}_{\kappa ,\chi }\left(1\right)\le 2(1-\varrho ).\hfill \end{array}$$

(38)

Proof. 

It is easy to show that, given Lemma 4 and (13),

$$\sum _{\iota =2}^{\infty }[(\iota -1)(\varpi (\iota +1)-1)+\iota (1-\varrho )][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\le 1-\varrho .$$

(39)

Since $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$, then $|c_{\iota }|\le \iota$. Inequality (39) can be represented as

$$\begin{array}{ccc}& & \sum _{\iota =2}^{\infty }[(\iota -1)(\varpi (\iota +1)-1)+\iota (1-\varrho )][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\hfill \\ & \le & \sum _{\iota =2}^{\infty }\iota [(\iota -1)(\varpi (\iota +1)-1)+\iota (1-\varrho )][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }\left[\varpi \vartheta \iota (\iota -1)(\iota -2)(\iota -3)+\left[\varpi \right(1+5\vartheta )-\vartheta \varrho ]\iota (\iota -1)(\iota -2)\right.\hfill \\ & +& \left.\left[\right(1+2\vartheta \left)\right(2\varpi -\varrho )+(1-\varpi \left)\right(\vartheta -1)+1]\iota (\iota -1)+(1-\varrho )\iota \right]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \varpi \vartheta \sum _{\iota =4}^{\infty }{\displaystyle \frac{\iota (\iota -1)(\iota -2)(\iota -3)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+[\varpi (1+5\vartheta )-\vartheta \varrho ]\sum _{\iota =3}^{\infty }{\displaystyle \frac{\iota (\iota -1)(\iota -2)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & +& [(1+2\vartheta )(2\varpi -\varrho )+(1-\varpi )(\vartheta -1)+1]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & +& (1-\varrho )\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}.\hfill \end{array}$$

Using (15)–(18), we thus get

$$\begin{array}{ccc}\hfill \varpi \vartheta {\mathbb{M}}_{\kappa ,\chi }^{\left(4\right)}\left(1\right)& +& [\varpi (1+5\vartheta )-\vartheta \varrho ]{\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)\hfill \\ & +& [(1+2\vartheta )(2\varpi -\varrho )+(1-\varpi )(\vartheta -1)+1]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+(1-\varrho )[{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]\le 1-\varrho .\hfill \end{array}$$

As a result, Theorem 11 is proved. □

Applying Theorem 11 and $\vartheta =0$, we get the next outcome:

Corollary 15. 

The function ${\mathfrak{I}}_{\chi }^{\kappa }\hslash \left(\zeta \right),$ defined by (10), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\widehat{\mathfrak{F}}(\varpi ,\varrho )$ if and only if

$$\begin{array}{c}\hfill \varpi {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[3\varpi -\varrho ]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+(1-\varrho ){\mathbb{M}}_{\kappa ,\chi }\left(1\right)\le 2(1-\varrho ).\end{array}$$

Applying Theorem 11 and $\varpi =1$, we get the next outcome:

Corollary 16. 

The function ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right),$ defined by (13), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\widehat{\mathcal{K}}\left(\varrho \right)$ if and only if

$$\begin{array}{c}\hfill \vartheta {\mathbb{M}}_{\kappa ,\chi }^{\left(4\right)}\left(1\right)+[1+\vartheta (5-\varrho )]{\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+[(1+2\vartheta )(2-\varrho )+1]{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+(1-\varrho ){\mathbb{M}}_{\kappa ,\chi }\left(1\right)\le 2(1-\varrho ).\end{array}$$

Taking $\vartheta =\varrho =0$ in Corollary 16, we get the next result:

Example 8. 

The function ${\mathfrak{I}}_{\chi }^{\kappa }\hslash \left(\zeta \right),$ defined by (10), maps $\hslash \left(\zeta \right)\in {\widehat{\mathcal{S}}}^{*}$ to $\widehat{\mathcal{K}}$ if and only if

$$\begin{array}{c}\hfill {\mathbb{M}}_{\kappa ,\chi }^{‴}\left(1\right)+3{\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+{\mathbb{M}}_{\kappa ,\chi }\left(1\right)\le 2.\end{array}$$

Theorem 12. 

The function ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right),$ defined by (13), maps $\hslash \left(\zeta \right)\in \widehat{\mathcal{K}}$ to $\widehat{\mathfrak{F}}(\varpi ,\varrho )$ if and only if (33) is satisfied.

Proof. 

According to Lemma 4 and (13), we must demonstrate that

$$\sum _{\iota =2}^{\infty }[(\iota -1)(\varpi (\iota +1)-1)+\iota (1-\varrho )][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\le 1-\varrho .$$

Since $\hslash \left(\zeta \right)\in \widehat{\mathcal{K}}$, then $|c_{\iota }|\le 1.$ Since Theorem 12’s proof has similarities to that of Theorem 10, we omit it. □

3. Properties of an Integral Operator

In this section, we examine the geometric features of the Mittag-Leffler function ${\mathbb{M}}_{\kappa ,\chi }\left(\zeta \right)$ under the action of the integral operator

$$\begin{array}{ccc}\hfill {\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right)& =& {\int }_0^{\zeta }\left[2-{\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}\right]ds\hfill \\ & =& \zeta -\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}{\zeta }^{\iota }.\hfill \end{array}$$

(40)

We aim to establish sufficient conditions under which the function ${\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right)$ belongs to the subclasses $\tilde{\Xi }(\varpi ,\varrho )$, $\widehat{\mathfrak{L}}(\varpi ,\varrho )$, $\widehat{\mathfrak{K}}(\varpi ,\varrho )$ and $\widehat{\mathfrak{F}}(\varpi ,\varrho )$.

Theorem 13. 

The function ${\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right),$ given by (40), is in $\tilde{\Xi }(\varpi ,\varrho )$ if and only if

$$\varpi {\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+(1-\varpi ){\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1\le 1-\varrho .$$

Proof. 

As stated in Lemma 1, we have to prove if

$$\sum _{\iota =2}^{\infty }\left[\iota (1+\varpi (\iota -1))\right]{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\le 1-\varrho .$$

(41)

Considering that the left side of (41) might be written as

$$\begin{array}{ccc}\hfill {\Phi }_1(\varpi ,\kappa ,\chi )& =& \sum _{\iota =2}^{\infty }\left[\iota (1+\varpi (\iota -1))\right]{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }[1+\varpi (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \varpi \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+(1-\varpi )\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}.\hfill \end{array}$$

Therefore, using (14) and (15), we get

$$\begin{array}{ccc}\hfill {\Phi }_1(\varpi ,\kappa ,\chi )& =& \varpi [{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]+(1-\varpi )[{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1].\hfill \end{array}$$

The conclusion that follows is that (41) is true. As a result, Theorem 13 is proved. □

Taking $\varpi =0$ in Theorem 13, we get the next result:

Corollary 17. 

The function ${\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right),$ given by (40), is in $\widehat{\mathcal{C}}\left(\varrho \right)$ if and only if

$${\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1\le 1-\varrho .$$

(42)

Theorem 14. 

The function ${\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right),$ given by (40), is in $\widehat{\mathfrak{L}}(\varpi ,\varrho )$ if and only if

$$\varpi {\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+(1-\varpi ){\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1-\varrho {\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\le 1-\varrho .$$

(43)

Proof. 

As stated in Lemma 2, we have to prove if

$$\sum _{\iota =2}^{\infty }[(\iota -1)(\varpi \iota +1)+1-\varrho ]{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\le 1-\varrho .$$

(44)

The left-hand side of (44) might be represented as

$$\begin{array}{ccc}\hfill {\Phi }_2(\varpi ,\varrho ,\kappa ,\chi )& =& \sum _{\iota =2}^{\infty }[(\iota -1)(\varpi \iota +1)+1-\varrho ]{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }\left[\varpi \iota (\iota -1)+(\iota -1)+(1-\varrho )\right]{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }\left[\varpi \iota +(1-\varpi )-{\displaystyle \frac{\varrho }{\iota }}\right]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \varpi \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+(1-\varpi )\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}-\varrho \sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}.\hfill \end{array}$$

Consequently, using (14), (15) and (19), we get

$$\begin{array}{ccc}\hfill {\Phi }_2(\varpi ,\varrho ,\kappa ,\chi )& =& \varpi [{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]+(1-\varpi )[{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1]-\varrho {\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds.\hfill \end{array}$$

Thus, it may be concluded that (44) is true. □

Taking $\varpi =0$ in Theorem 14, we get the next outcome:

Corollary 18. 

The function ${\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right),$ defined by (40), is in ${\widehat{\mathcal{S}}}^{*}\left(\varrho \right)$ if and only if

$${\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1-\varrho {\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\le 1-\varrho .$$

Taking $\varrho =0$ in Corollary 18, we get the next outcome:

Example 9. 

The function ${\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right),$ defined by (40), is in ${\widehat{\mathcal{S}}}^{*}$ if and only if

$${\mathbb{M}}_{\kappa ,\chi }\left(1\right)\le 2.$$

Theorem 15. 

The function ${\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right),$ defined by (40), is in $\widehat{\mathfrak{K}}(\varpi ,\varrho )$ if and only if

$$\varpi {\mathbb{M}}_{\kappa ,\chi }\left(1\right)-\varpi +(1-\varpi ){\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\le 1-\varrho .$$

Proof. 

As stated in Lemma 3, we have to prove if

$$\sum _{\iota =2}^{\infty }[1+\varpi (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\le 1-\varrho .$$

(45)

Considering that the left side of (45) might be represented as

$$\begin{array}{ccc}\hfill {\Phi }_3(\varpi ,\kappa ,\chi )& =& \sum _{\iota =2}^{\infty }[1+\varpi (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }\left[\varpi +{\displaystyle \frac{1-\varpi }{\iota }}\right]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \varpi \sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+(1-\varpi )\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}.\hfill \end{array}$$

Therefore, using (14) and (19), we get

$$\begin{array}{ccc}\hfill {\Phi }_3(\varpi ,\kappa ,\chi )& =& \varpi [{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1]+(1-\varpi ){\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds.\hfill \end{array}$$

The conclusion that follows is that (45) is true. As a result, Theorem 15 is proved. □

Taking $\varpi =1$ in Theorem 15, we get the next outcome:

Corollary 19. 

The function ${\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right),$ given by (40), is in $\widehat{\mathcal{C}}\left(\varrho \right)$ if and only if (42) satisfies.

Theorem 16. 

The function ${\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right),$ given by (40), is in $\widehat{\mathfrak{F}}(\varpi ,\varrho )$ if and only if

$$\varpi {\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-\varrho {\mathbb{M}}_{\kappa ,\chi }\left(1\right)+(\varrho -\varpi )+(1-\varpi ){\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\le 1-\varrho .$$

Proof. 

As stated in Lemma 4, we have to prove if

$$\sum _{\iota =2}^{\infty }[(\iota -1)(\varpi (\iota +1)-1)+\iota (1-\varrho )]{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\le 1-\varrho .$$

(46)

Considering that the left-hand side of (46) might be represented as

$$\begin{array}{ccc}\hfill {\Phi }_4(\varpi ,\varrho ,\kappa ,\chi )& =& \sum _{\iota =2}^{\infty }[(\iota -1)(\varpi (\iota +1)-1)+\iota (1-\varrho )]{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \sum _{\iota =2}^{\infty }\left[\iota \varpi -\varrho +{\displaystyle \frac{1-\varpi }{\iota }}\right]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& \varpi \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}-\varrho \sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1\left)\right)+\chi )}}+(1-\varpi )\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}.\hfill \end{array}$$

Therefore, using (14), (15) and (19), we get

$$\begin{array}{ccc}\hfill {\Phi }_4(\varpi ,\varrho ,\kappa ,\chi )& =& \varpi [{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]-\varrho [{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1]+(1-\varpi ){\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds.\hfill \end{array}$$

The conclusion that follows is that (46) is true. As a result, Theorem 16 is proved. □

Taking $\varpi =1$ in Theorem 16, we get the next outcome:

Corollary 20. 

The function ${\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right),$ given by (40), is in $\widehat{\mathcal{K}}\left(\varrho \right)$ if and only if

$${\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-\varrho {\mathbb{M}}_{\kappa ,\chi }\left(1\right)+(\varrho -1)\le 1-\varrho .$$

Taking $\varrho =0$ in Corollary 20, we get the next result:

Example 10. 

The function ${\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right),$ given by (40), is in $\widehat{\mathcal{K}}$ if and only if

$${\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)\le 2.$$

4. Image Properties of ${\mathbb{W}}_{\mathit{\chi },\mathit{\vartheta }}^{\mathit{\kappa }}\mathit{h}\left(\mathit{\zeta }\right)$ Operator

In this section, we study the effect of the linear convolution operator ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right)$, defined using the Mittag-Leffler function, on functions belonging to the class ${\mathfrak{R}}_{\theta }(E,V)$. We derive sufficient conditions under which the transformed functions ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right)$ belong to the subclasses $\Xi (\varpi ,\varrho ),\mathfrak{L}(\varpi ,\varrho ),\mathfrak{K}(\varpi ,\varrho )$ and $\mathfrak{F}(\varpi ,\varrho )$.

Theorem 17. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right),$ given by (11). The operator ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \Xi (\varpi ,\varrho )$ if

$$(E-V)\left|\theta \right|[\vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[(1-\varpi )\vartheta +\varpi ]{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+(1-\varpi )(1-\vartheta ){\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1]\le 1-\varrho .$$

(47)

Proof. 

Since $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, Lemma 1 and (11) state that we must demonstrate that

$$\sum _{\iota =2}^{\infty }\left[\iota (1+\varpi (\iota -1))\right][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\le 1-\varrho .$$

(48)

Applying Lemma 5 to (48), we can represent it as

$$\begin{array}{ccc}& & \sum _{\iota =2}^{\infty }\left[\iota (1+\varpi (\iota -1))\right][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\hfill \\ & \le & \sum _{\iota =2}^{\infty }\left[\iota (1+\varpi (\iota -1))\right][1+\vartheta (\iota -1)]\left({\displaystyle \frac{(E-V)\left|\theta \right|}{\iota }}\right){\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& (E-V)\left|\theta \right|\left[\sum _{\iota =2}^{\infty }[\vartheta \varpi \iota (\iota -1)+[(1-\varpi )\vartheta +\varpi )\iota +(1-\varpi )(1-\vartheta )]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\right]\hfill \\ & =& (E-V)\left|\theta \right|\left[\vartheta \varpi \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+[(1-\varpi )\vartheta +\varpi ]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\right.\hfill \\ & +& \left.(1-\varpi )(1-\vartheta )\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\right].\hfill \end{array}$$

Thus, with (14)–(16), we obtain

$$\begin{array}{c}\hfill (E-V)\left|\theta \right|[\vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[(1-\varpi )\vartheta +\varpi ][{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]+(1-\varpi )(1-\vartheta )[{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1]]\le 1-\varrho .\end{array}$$

As a result, Theorem 17 is proved. □

Taking $\vartheta =0$ in Theorem 17, we get the following case:

Corollary 21. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{H}}_{\chi }^{\kappa }h\left(\zeta \right)$ given by (8). The operator ${\mathbb{H}}_{\chi }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \Xi (\varpi ,\varrho )$ if

$$(E-V)\left|\theta \right|\left[\varpi {\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+(1-\varpi ){\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1\right]\le 1-\varrho .$$

For $\varpi =0$ in Theorem 17, we get the following case:

Corollary 22. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right)$ given by (11). The operator ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathcal{C}\left(\varrho \right)$ if

$$(E-V)\left|\theta \right|\left[\vartheta {\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+(1-\vartheta ){\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1\right]\le 1-\varrho .$$

(49)

Theorem 18. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right)$ given by (11). The operator ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathfrak{L}(\varpi ,\varrho )$ if

$$\begin{array}{cc}\hfill (E-V)\left|\theta \right|& [\vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[(1-\varpi )\vartheta +\varpi ]{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+[1-\varpi +\vartheta (\varpi -\varrho -1)]{\mathbb{M}}_{\kappa ,\chi }\left(1\right)\hfill \\ \hfill & \left.+(\vartheta \varrho -1)+\varrho (\vartheta -1){\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .\hfill \end{array}$$

Proof. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V).$ Given Lemma 2 and (11), it is straightforward to demonstrate that

$$\sum _{\iota =2}^{\infty }[(\iota -1)(\varpi \iota +1)+1-\varrho ][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\le 1-\varrho .$$

(50)

Applying Lemma 5 to (50), we can represent it as

$$\begin{array}{ccc}& & \sum _{\iota =2}^{\infty }[(\iota -1)(\varpi \iota +1)+1-\varrho ][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\hfill \\ & \le & \sum _{\iota =2}^{\infty }[(\iota -1)(\varpi \iota +1)+1-\varrho ][1+\vartheta (\iota -1)]\left({\displaystyle \frac{(E-V)\left|\theta \right|}{\iota }}\right){\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& (E-V)\left|\theta \right|[\sum _{\iota =2}^{\infty }\left[\varpi \vartheta \iota {(\iota -1)}^2+\vartheta {(\iota -1)}^2+\varpi \iota (\iota -1)+[(1-\varrho )\vartheta +1](\iota -1)\right.\hfill \\ & +& \left.(1-\varrho )]{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\right]\hfill \\ & =& (E-V)\left|\theta \right|\left[\vartheta \varpi \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+[\varpi +\vartheta (1-\varpi )]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\right.\hfill \\ & +& \left.[1-\varpi +\vartheta (\varpi -\varrho -1)]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+\varrho (\vartheta -1)\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\right]\hfill \end{array}$$

Thus, with (14)–(16) and (19), we obtain

$$\begin{array}{ccc}\hfill (E& -& V\left)\right|\theta \left|\right[\vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[\varpi +\vartheta (1-\varpi )][{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]\hfill \\ & +& \left.[1-\varpi +\vartheta (\varpi -\varrho -1)][{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1]+\varrho (\vartheta -1){\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .\hfill \end{array}$$

As a result, Theorem 18 is proved. □

Taking $\vartheta =0$ in Theorem 18, we get the following case:

Corollary 23. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{H}}_{\chi }^{\kappa }h\left(\zeta \right)$ given by (8). The operator ${\mathbb{H}}_{\chi }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathfrak{L}(\varpi ,\varrho )$ if

$$(E-V)\left|\theta \right|\left[\varpi {\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+[1-\varpi ]{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1-\varrho {\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .$$

For $\varpi =0$ in Theorem 18, we get the following case:

Corollary 24. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right)$ given by (11). The operator ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset {\mathcal{S}}^{*}\left(\varrho \right)$ if

$$\begin{array}{cc}\hfill (E-V)\left|\theta \right|& [\vartheta {\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+[1-\vartheta (\varrho +1)]{\mathbb{M}}_{\kappa ,\chi }\left(1\right)\hfill \\ \hfill & \left.+(\vartheta \varrho -1)+\varrho (\vartheta -1){\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .\hfill \end{array}$$

For $\vartheta =\varrho =0$ in Corollary 24, we get the following case:

Example 11. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{H}}_{\chi }^{\kappa }h\left(\zeta \right)$ given by (8). The operator ${\mathbb{H}}_{\chi }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset {\mathcal{S}}^{*}$ if

$$(E-V)\left|\theta \right|\left[{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1\right]\le 1.$$

Theorem 19. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right)$ given by (11). The operator ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathfrak{K}(\varpi ,\varrho )$ if

$$\begin{array}{cc}\hfill (E-V)\left|\theta \right|& [\vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+[\varpi (1-2\vartheta )+\vartheta ]{\mathbb{M}}_{\kappa ,\chi }\left(1\right)+[(\varpi -1)\vartheta -\varpi ]\hfill \\ & \left.+(1-\varpi )(1-\vartheta ){\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .\hfill \end{array}$$

Proof. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$. It is easy to show that, given Lemma 3 and (11),

$$\sum _{\iota =2}^{\infty }[(\iota -1)\varpi +1][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\le 1-\varrho .$$

(51)

Applying Lemma 5 to (51), we can represent it as

$$\begin{array}{ccc}& & \sum _{\iota =2}^{\infty }[(\iota -1)\varpi +1][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\hfill \\ & \le & \sum _{\iota =2}^{\infty }[\varpi (\iota -1)+1][1+\vartheta (\iota -1)]\left({\displaystyle \frac{(E-V)\left|\theta \right|}{\iota }}\right){\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& (E-V)\left|\theta \right|\sum _{\iota =2}^{\infty }\left[\varpi \vartheta \iota +[\vartheta +\varpi (1-2\vartheta )]+{\displaystyle \frac{(1-\vartheta )(1-\varpi )}{\iota }}\right]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& (E-V)\left|\theta \right|\left[\varpi \vartheta \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+[\vartheta +\varpi (1-2\vartheta )]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\right.\hfill \\ & +& \left.(1-\varpi )(1-\vartheta ){\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\right].\hfill \end{array}$$

Using (14), (15) and (19), we thus get

$$\begin{array}{ccc}\hfill (E-V)\left|\theta \right|\left[\varpi \vartheta \right[{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]& +& [\vartheta +\varpi (1-2\vartheta )][{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1]\hfill \\ & +& \left.(1-\varpi )(1-\vartheta ){\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .\hfill \end{array}$$

As a result, Theorem 19 is proved. □

Taking $\vartheta =0$ in Theorem 19, we get the following case:

Corollary 25. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{H}}_{\chi }^{\kappa }h\left(\zeta \right)$ given by (8). The operator ${\mathbb{H}}_{\chi }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathfrak{K}(\varpi ,\varrho )$ if

$$(E-V)\left|\theta \right|\left[\varpi {\mathbb{M}}_{\kappa ,\chi }\left(1\right)-\varpi +(1-\varpi ){\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .$$

For $\varpi =1$ in Theorem 19, we get the following case:

Corollary 26. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right)$ given by (11). The operator ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathcal{C}\left(\varrho \right)$ if (49) satisfies.

Theorem 20. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right)$ given by (11). The operator ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathfrak{F}(\varpi ,\varrho )$ if

$$\begin{array}{cc}\hfill (E-V)\left|\theta \right|& [\vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[\varpi -\vartheta \varrho ]{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+[\vartheta (\varrho -\varpi +1)-\varrho ]{\mathbb{M}}_{\kappa ,\chi }\left(1\right)\hfill \\ \hfill & \left.+[\varrho -\varpi +\vartheta (\varpi -1)]+(1-\varpi )(1-\vartheta ){\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .\hfill \end{array}$$

Proof. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$. It is easy to show that, given Lemma 4 and (11),

$$\sum _{\iota =2}^{\infty }[(\iota -1)(\varpi (\iota +1)-1)+\iota (1-\varrho )][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\le 1-\varrho .$$

(52)

Applying Lemma 5 to (52), we can represent it as

$$\begin{array}{ccc}& & \sum _{\iota =2}^{\infty }[(\iota -1)(\varpi (\iota +1)-1)+\iota (1-\varrho )][1+\vartheta (\iota -1)]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\left|c_{\iota }\right|\hfill \\ & \le & \sum _{\iota =2}^{\infty }[(\iota -1)(\varpi (\iota +1)-1)+\iota (1-\varrho )][1+\vartheta (\iota -1)]\left({\displaystyle \frac{(E-V)\left|\theta \right|}{\iota }}\right){\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\hfill \\ & =& (E-V)\left|\theta \right|\left[\sum _{\iota =2}^{\infty }[\vartheta \varpi \iota (\iota -1)+[\varpi -\vartheta \varrho ]\iota +[\vartheta (\varrho -\varpi +1)-\varrho ]\right.\hfill \\ & +& \left.\left.{\displaystyle \frac{(1-\varpi )(1-\vartheta )}{\iota }}\right]{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\right]\hfill \\ & =& (E-V)\left|\theta \right|\left[\vartheta \varpi \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+[\varpi -\vartheta \varrho ]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota \Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}\right.\hfill \\ & +& \left.[\vartheta (\varrho -\varpi +1)-\varrho ]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\Gamma \left(\kappa \right(\iota -1)+\chi )}}+\left[(1-\varpi )(1-\vartheta )\right]\sum _{\iota =2}^{\infty }{\displaystyle \frac{\Gamma \left(\chi \right)}{\iota \Gamma \left(\kappa \right(\iota -1)+\chi )}}\right].\hfill \end{array}$$

Using (14)–(16) and (19), we thus get

$$\begin{array}{ccc}& & (E-V)\left|\theta \right|[\vartheta \varpi {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[\varpi -\vartheta \varrho ][{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]\hfill \\ & +& \left.[\vartheta (\varrho -\varpi +1)-\varrho ][{\mathbb{M}}_{\kappa ,\chi }\left(1\right)-1]+\left[(1-\varpi )(1-\vartheta )\right]{\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .\hfill \end{array}$$

As a result, Theorem 20 is proved. □

Taking $\vartheta =0$ in Theorem 20, we get the following case:

Corollary 27. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{H}}_{\chi }^{\kappa }h\left(\zeta \right)$ given by (8). The operator ${\mathbb{H}}_{\chi }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathfrak{F}(\varpi ,\varrho )$ if

$$(E-V)\left|\theta \right|\left[\varpi {\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-\varrho {\mathbb{M}}_{\kappa ,\chi }\left(1\right)+[\varrho -\varpi ]+(1-\varpi ){\int }_0^1\left({\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .$$

For $\varpi =1$ in Theorem 20, we get the following case:

Corollary 28. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right)$ given by (11). The operator ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathcal{K}\left(\varrho \right)$ if

$$(E-V)\left|\theta \right|[\vartheta {\mathbb{M}}_{\kappa ,\chi }^{″}\left(1\right)+[1-\vartheta \varrho ]{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)+\left[\varrho (\vartheta -1)\right]{\mathbb{M}}_{\kappa ,\chi }\left(1\right)+\varrho -1]\le 1-\varrho .$$

For $\vartheta =\varrho =0$ in Corollary 28, we get the following case:

Example 12. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, ${\mathbb{H}}_{\chi }^{\kappa }h\left(\zeta \right)$ given by (8). The operator ${\mathbb{H}}_{\chi }^{\kappa }\left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathcal{K}$ if

$$(E-V)\left|\theta \right|[{\mathbb{M}}_{\kappa ,\chi }^{\prime }\left(1\right)-1]\le 1.$$

5. Some Examples and Consequences

For the special case $\kappa =\chi =1$, the normalized Mittag-Leffler function is given by

$$\begin{array}{ccc}\hfill \lambda \left(\zeta \right)={\mathbb{M}}_{1,1}\left(\zeta \right)& =& \zeta +\sum _{\iota =2}^{\infty }{\displaystyle \frac{1}{\Gamma \left(\iota \right)}}{\zeta }^{\iota },\lambda \left(1\right)-1=\sum _{\iota =2}^{\infty }{\displaystyle \frac{1}{\Gamma \left(\iota \right)}},\hfill \\ \hfill {\lambda }^{\prime }\left(1\right)-1& =& \sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota }{\Gamma \left(\iota \right)}},{\lambda }^{″}\left(1\right)=\sum _{\iota =2}^{\infty }{\displaystyle \frac{\iota (\iota -1)}{\Gamma \left(\iota \right)}},\hfill \\ \hfill {\int }_0^1\left({\displaystyle \frac{\lambda \left(s\right)}{s}}-1\right)ds& =& \sum _{\iota =2}^{\infty }{\displaystyle \frac{1}{\iota \Gamma \left(\iota \right)}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \mathcal{E}\left(\zeta \right)& =& \zeta -\sum _{\iota =2}^{\infty }{\displaystyle \frac{1}{\iota \Gamma \left(\iota \right)}}{\zeta }^{\iota }\hfill \end{array}$$

(53)

$$\begin{array}{ccc}\hfill \mathfrak{P}(\vartheta ;\zeta )& =& \zeta +\sum _{\iota =2}^{\infty }[1+\vartheta (\iota -1)]{\displaystyle \frac{c_{\iota }}{\Gamma \left(\iota \right)}}{\zeta }^{\iota }.\hfill \end{array}$$

(54)

Using the above relations in Theorem 13, we get the following case:

Corollary 29. 

The function $\mathcal{E}\left(\zeta \right),$ defined by (53) is in $\tilde{\Xi }(\varpi ,\varrho )$ if and only if

$$\varpi {\lambda }^{\prime }\left(1\right)+(1-\varpi )\lambda \left(1\right)-1\le 1-\varrho .$$

For $\varpi =0$ in Corollary 29, we obtain the following case:

Example 13. 

The function $\mathcal{E}\left(\zeta \right),$ defined by (53), is in $\mathcal{C}\left(\varrho \right)$ if and only if

$$\lambda \left(1\right)-1\le 1-\varrho .$$

Using the above relations in Theorem 14, we get the following case:

Corollary 30. 

The function $\mathcal{E}\left(\zeta \right),$ defined by (53), is in $\widehat{\mathfrak{L}}(\varpi ,\varrho )$ if and only if

$$\varpi {\lambda }^{\prime }\left(1\right)+(1-\varpi )\lambda \left(1\right)-1-\varrho {\int }_0^1\left({\displaystyle \frac{\lambda \left(s\right)}{s}}-1\right)ds\le 1-\varrho .$$

For $\varpi =\varrho =0$ in Corollary 30, we obtain the following case:

Example 14. 

The function $\mathcal{E}\left(\zeta \right),$ defined by (53), is in ${\widehat{\mathcal{S}}}^{*}$ if and only if

$$\lambda \left(1\right)\le 2.$$

Using the above relations in Theorem 15, we get the following case:

Corollary 31. 

The function $\mathcal{E}\left(\zeta \right),$ defined by (53), is in $\widehat{\mathfrak{K}}(\varpi ,\varrho )$ if and only if

$$\varpi \lambda \left(1\right)-\varpi +(1-\varpi ){\int }_0^1\left({\displaystyle \frac{\lambda \left(s\right)}{s}}-1\right)ds\le 1-\varrho .$$

For $\varpi =1$ in Corollary 31, we obtain the following case:

Example 15. 

The function $\mathcal{E}\left(\zeta \right),$ defined by (53), is in $\mathcal{C}\left(\varrho \right)$ if and only if

$$\lambda \left(1\right)-1\le 1-\varrho .$$

Using the above relations in Theorem 16, we get the following case:

Corollary 32. 

The function $\mathcal{E}\left(\zeta \right),$ defined by (53), is in $\widehat{\mathfrak{F}}(\varpi ,\varrho )$ if and only if

$$\varpi {\lambda }^{\prime }\left(1\right)-\varrho \lambda \left(1\right)+(\varrho -\varpi )+(1-\varpi ){\int }_0^1\left({\displaystyle \frac{\lambda \left(s\right)}{s}}-1\right)ds\le 1-\varrho .$$

For $\varpi =1$ and $\varrho =0$ in Corollary 32, we obtain the following case:

Example 16. 

The function $\mathcal{E}\left(\zeta \right),$ defined by (53), is in $\widehat{\mathcal{K}}$ if and only if

$${\lambda }^{\prime }\left(1\right)\le 2.$$

Using the above relations in Theorem 17, we get the following case:

Corollary 33. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, with $\mathfrak{P}(\vartheta ;\zeta )$ defined by (54). The operator $\lambda \left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \Xi (\varpi ,\varrho )$ if

$$(E-V)\left|\theta \right|\left[\vartheta \varpi {\lambda }^{″}\left(1\right)+[(1-\varpi )\vartheta +\varpi ]{\lambda }^{\prime }\left(1\right)+(1-\varpi )(1-\vartheta )\lambda \left(1\right)-1\right]\le 1-\varrho .$$

For $\varpi =0$ in Corollary 33, we obtain the following case:

Example 17. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, $\mathfrak{P}(\vartheta ;\zeta )$ defined by (54). The operator $\lambda \left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathcal{C}\left(\varrho \right)$ if

$$(E-V)\left|\theta \right|\left[\vartheta {\lambda }^{\prime }\left(1\right)+(1-\vartheta )\lambda \left(1\right)-1\right]\le 1-\varrho .$$

Using the above relations in Theorem 18, we get the following case:

Corollary 34. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, $\mathfrak{P}(\vartheta ;\zeta )$ defined by (54). The operator $\lambda \left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathfrak{L}(\varpi ,\varrho )$ if

$$\begin{array}{cc}\hfill (E-V)\left|\theta \right|& \left[\vartheta \varpi {\lambda }^{″}\left(1\right)+[(1-\varpi )\vartheta +\varpi ]{\lambda }^{\prime }\left(1\right)+[1-\varpi +\vartheta (\varpi -\varrho -1)]\lambda \left(1\right)\right.\hfill \\ \hfill & \left.+(\vartheta \varrho -1)+\varrho (\vartheta -1){\int }_0^1\left({\displaystyle \frac{\lambda \left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .\hfill \end{array}$$

For $\varpi =\varrho =0$ in Corollary 34, we obtain the following case:

Example 18. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, $\mathfrak{P}(\vartheta ;\zeta )$ defined by (54). The operator $\lambda \left({\mathfrak{R}}_{\theta }(E,V)\right)\subset {\mathcal{S}}^{*}$ if

$$\begin{array}{cc}\hfill (E-V)\left|\theta \right|& \left[\vartheta {\lambda }^{\prime }\left(1\right)+[1-\vartheta ]\lambda \left(1\right)-1\right]\le 1.\hfill \end{array}$$

Using the above relations in Theorem 19, we get the following case:

Corollary 35. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, $\mathfrak{P}(\vartheta ;\zeta )$ defined by (54). The operator $\lambda \left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathfrak{K}(\varpi ,\varrho )$ if

$$\begin{array}{cc}\hfill (E-V)\left|\theta \right|& \left[\vartheta \varpi {\lambda }^{\prime }\left(1\right)+[\varpi (1-2\vartheta )+\vartheta ]\lambda \left(1\right)+[(\varpi -1)\vartheta -\varpi ]\right.\hfill \\ & +\left.(1-\varpi )(1-\vartheta ){\int }_0^1\left({\displaystyle \frac{\lambda \left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .\hfill \end{array}$$

For $\varpi =1$ in Corollary 35, we obtain the following case:

Example 19. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, $\mathfrak{P}(\vartheta ;\zeta )$ defined by (54). The operator $\lambda \left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathcal{C}\left(\varrho \right)$ if

$$(E-V)\left|\theta \right|\left[\vartheta {\lambda }^{\prime }\left(1\right)+[1-\vartheta ]\lambda \left(1\right)-1\right]\le 1-\varrho .$$

Using the above relations in Theorem 20, we get the following case:

Corollary 36. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, $\mathfrak{P}(\vartheta ;\zeta )$ defined by (54). The operator $\lambda \left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathfrak{F}(\varpi ,\varrho )$ if

$$\begin{array}{cc}\hfill (E-V)\left|\theta \right|\left[\vartheta \varpi \right.& {\lambda }^{″}\left(1\right)+[\varpi -\vartheta \varrho ]{\lambda }^{\prime }\left(1\right)+[\vartheta (\varrho +1)-\varrho ]\lambda \left(1\right)\hfill \\ \hfill & \left.+[\varrho -\varpi -\vartheta ]+[1-\varpi -\vartheta ]{\int }_0^1\left({\displaystyle \frac{\lambda \left(s\right)}{s}}-1\right)ds\right]\le 1-\varrho .\hfill \end{array}$$

For $\varpi =1$ and $\varrho =0$ in Corollary 36, we obtain the following case:

Example 20. 

Let $h\left(\zeta \right)\in {\mathfrak{R}}_{\theta }(E,V)$, $\mathfrak{P}(\vartheta ;\zeta )$ defined by (54). The operator $\lambda \left({\mathfrak{R}}_{\theta }(E,V)\right)\subset \mathcal{K}$ if

$$\begin{array}{c}\hfill (E-V)\left|\theta \right|\left[\vartheta {\lambda }^{″}\left(1\right)+{\lambda }^{\prime }\left(1\right)+\vartheta \lambda \left(1\right)-[1+\vartheta ]-\vartheta {\int }_0^1\left({\displaystyle \frac{\lambda \left(s\right)}{s}}-1\right)ds\right]\le 1.\end{array}$$

6. Conclusions

In this paper, we have undertaken a comprehensive study of certain subclasses of analytic functions defined within the unit disk, particularly those associated with the Mittag-Leffler function ${\mathbb{M}}_{\kappa ,\chi }\left(\zeta \right)$. We began by employing convolution (Hadamard product) techniques to define a new linear operator ${\mathfrak{S}}_{\chi ,\vartheta }^{\kappa }\hslash \left(\zeta \right)$, which maps analytic functions to new functions with transformed coefficients derived from the Mittag-Leffler function. We established new sufficient conditions to ensure that functions related to the generalized Mittag-Leffler function are members of analytic function subclasses, namely $\tilde{\Xi }(\varpi ,\varrho ),\widehat{\mathfrak{L}}(\varpi ,\varrho ),\widehat{\mathfrak{K}}(\varpi ,\varrho )$, and $\widehat{\mathfrak{F}}(\varpi ,\varrho )$. As a generalization of the standard classes of close-to-convex, convex, and starlike functions, we study inclusion outcomes in the recently described subclasses $\tilde{\Xi }(\varpi ,\varrho ),\widehat{\mathfrak{L}}(\varpi ,\varrho ),\widehat{\mathfrak{K}}(\varpi ,\varrho )$, and $\widehat{\mathfrak{F}}(\varpi ,\varrho )$. In addition to convolution, we analyzed the effect of the integral operator ${\mathfrak{Z}}_{\chi }^{\kappa }\left(\zeta \right)={\int }_0^{\zeta }\left[2-{\displaystyle \frac{{\mathbb{M}}_{\kappa ,\chi }\left(s\right)}{s}}\right]ds$, which serves the Mittag-Leffler function integral transformation. We derived sufficient conditions for this operator’s image to belong to both $\tilde{\Xi }(\varpi ,\varrho ),\widehat{\mathfrak{L}}(\varpi ,\varrho ),\widehat{\mathfrak{K}}(\varpi ,\varrho )$, and $\widehat{\mathfrak{F}}(\varpi ,\varrho )$, demonstrating that even under integration, the essential geometric properties can be preserved under careful parameter selection. Also, we examine the behavior of functions from ${\mathfrak{R}}_{\theta }(E,V)$ under the action of the convolution operator ${\mathbb{W}}_{\chi ,\vartheta }^{\kappa }h\left(\zeta \right)$, establishing sufficient criteria for the resulting images to lie within the subclasses of the analytic function. We gave specific instances to support our theoretical conclusions. Mittag-Leffler functions arise naturally as solutions of fractional differential equations and in models of anomalous diffusion and viscoelasticity. The operators and subclasses studied here provide analytic tools that may be applied to investigate stability and qualitative behavior in such systems, thereby linking our theoretical results with practical applications in fractional calculus. Some of the inequalities obtained here are sharp, with extremal functions arising from well-known starlike, convex and close-to-convex families; for example, Example 1 refers to the condition for Mittag-Leffler belonging to the class of convex functions. Finally, the methods developed in this work open several avenues for future research. Possible extensions include the study of analogous operators acting on multivalent and bi-univalent function classes, as well as further exploration of connections with fractional dynamical systems and anomalous diffusion models. These directions highlight the broader significance and potential applicability of the results presented in this study.