Geometric Studies on Mittag-Leffler Type Function Involving a New Integrodifferential Operator

Abstract

The generalized exponential function in a complex domain is called the Mittag-Leffler function (MLF). The implementations of MLF are significant in diverse areas of science. Over the past few decades, MLF and its analysis with generalizations have become an increasingly rich research area in mathematics and its allied fields. In the geometric theory of meromorphic functions, the main contribution to this discipline of study is to enrich areas of operator theory on complex punctured domains and differential complex inequalities, namely, subordination theory. This effort presents integrodifferential operator of meromorphic functions in the punctured unit disk. It is formulated by combining the differential operator and the integral operator correlating with the extended generalized Mittag-Leffler function. Furthermore, some interesting geometric features in terms of the subordination principle are investigated.

1. Introduction

Complex analysis (complex function theory) began in the 18th century and has since become one of the significant topics in mathematics. Because of its efficient applicability to a wide range of concepts and problems, this domain has significantly impacted a wide range of research areas, including engineering, physics, and mathematics. Researchers discovered some unexpected connections between ostensibly disparate study fields. Mittag-Leffler function (M-LF) research is an unusual and fascinating combination of geometry and complex analysis that deals with the structure of analytic functions in the complex domain.

The Mittag-Leffler function (M-LF), which plays a vital role in a variety of issues in fractional calculus, operator theory, mathematical analysis, and other domains related to sciences and engineering, has been a topic that has inspired several researchers. As a result, researchers have become more interested in M-LF activity and, as a result, expanded their findings to the complex domain. M-LF is a fantastic tool used in geometric function theory (GFT) to consider multiple geometric properties of regulatory (analytical) functions and suggest a number of new operators. This function is a popularization extension of the exponential function. It was first proposed in 1903 [1] with a single parameter by Mittag-Leffler, a prominent mathematician:

$$\begin{array}{c}\hfill {\mathfrak{M}}_{\varrho }\left(z\right)=\sum _{\kappa =0}^{\infty }\frac{z^{\kappa }}{\Gamma (1+\varrho \kappa )},\left(z\in \mathbb{C},\Re \left\{\varrho \right\}>0\right).\end{array}$$

Due to extensive research in this field, the study of M-LF has emerged as one of the crucial types in special function theory (SFT). According to Wiman [2,3], the first generalized M-LF with two parameters was suggested in 1905 and was written as follows:

$$\begin{array}{c}\hfill {\mathfrak{M}}_{\tau ,\varrho }\left(z\right)=\sum _{\kappa =0}^{\infty }\frac{z^{\kappa }}{\Gamma (\tau +\varrho \kappa )},\left(z\in \mathbb{C},\Re \left(\varrho \right)>0,\tau \in \mathbb{C}\right).\end{array}$$

In addition to the expansions offered by Srivastava [4] in 1968, Prabhakar [5] in 1971 developed a formula for the generalized M-LF with three parameters, known as the Prabhakar function, as follows:

$$\begin{array}{c}\hfill {\mathfrak{M}}_{\tau ,\varrho }^a\left(z\right)=\sum _{\kappa =0}^{\infty }\frac{{\left(a\right)}_{\kappa }}{\Gamma (\varrho \kappa +\tau )}\frac{z^{\kappa }}{\kappa !},\left(z\in \mathbb{C},\Re \left(\varrho \right)>0,\tau ,a\in \mathbb{C}\right).\end{array}$$

Srivastava contributed substantially to investigating numerous M-LF features, extensions, generalizations, and implementations in this domain. According to Srivastava and Tomovski [6], who researched a further extension of M-LF in 2009, their primary contributions are as follows:

$$\begin{array}{c}\hfill {\mathfrak{M}}_{\tau ,\varrho }^a\left(z\right)=\sum _{\kappa =0}^{\infty }\frac{{\left(a\right)}_{\mathsf{n}\kappa }}{\Gamma (\varrho \kappa +\tau )}\frac{z^{\kappa }}{\kappa !},\left(z\in \mathbb{C},\Re \left(\varrho \right)>max\{0,\Re (\mathsf{n})-1\},\tau ,a\in \mathbb{C},\Re \left(\mathsf{n}\right)>0\right).\end{array}$$

Additionally, the authors looked at a more general fractional integral operator utilizing this broader class of M-LF. After that, Tomovskia et al. [7] in 2010 identified some compositional features that link with the specialization of the equivalent M-LF studied in [6]. A study of some recent developments covering several classes of the M-LF type related to diversified sorts of general Riemann–Liouville and other pertinent fractional-derivative-type operators was provided by Srivastava in 2016 [8]. Srivastava et al. [9] also implemented a new convolutional operator on the right-half of the complex domain formulated as an open disk when $Re\left(z\right)>0$ in the model of the generalized M-LF in terms of the well-researched Fox–Wright complex function. Employing Sumudu transform and Laplacian transform methods, Kumar et al. [10] (2018) investigated the generalized M-LF integration into resolving the generalized-kinetic-type fractional equation. The M-LF type with generalized multi-index was first developed by Saxena and Nishimoto [11] in 2010. Eight years later, Srivastava et al. [12] proposed a generic fractional-integral-type operator that encompasses it. Some scientists also found various findings that expanded the similar conclusions by Srivastava and Tomovski [6], and Kilbas et al. [13]. Using Riemann–Liouville fractional calculus, Srivastava et al. [14] recently presented significant relationships among the M-LF of the first, second, and third parameters. The investigation of M-LF and its different formulae has, in fact, drawn much attention, leading to the emergence of several scholars in this area. As an example, [15,16,17,18,19,20,21,22,23,24,25].

In connection with this line of research, Parmar (2015) [26] proposed a stunningly novel universal class of generalized M-LF extended as:

$$\begin{array}{c}\hfill {\mathfrak{M}}_{\tau ,\varrho }^{\left({\left\{{\mathit{a}}_{\mathit{j}}\right\}}_{j\in {\mathbb{N}}_0};\sigma \right)}(z;s)=\sum _{\kappa =0}^{\infty }\frac{{\mathcal{B}}^{\left({\left\{{\mathit{a}}_j\right\}}_{\mathit{j}\in {\mathbb{N}}_0}\right)}(\sigma +\kappa ,1-\sigma ;s)}{B(\sigma ,1-\sigma )}\frac{z^{\kappa }}{\Gamma (\varrho \kappa +\tau )},\end{array}$$

(1)

$$\left(z\in \mathbb{C};s\ge 0,\Re \left(\sigma \right)>0,\Re \left(\tau \right)>0,\Re \left(\varrho \right)>0;\varrho ,\tau ,\sigma \in \mathbb{C}\right),$$

where the complex plane is represented by $\mathbb{C}$, the classical beta function is represented by $B(\sigma ,1-\sigma )$, and ${\mathcal{B}}^{\left({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0}\right)}(\sigma +\kappa ,1-\sigma ;s)$ represents the classical extended beta function, constructed in [26], as follows:

$$\begin{array}{c}\hfill {\mathcal{B}}^{\left({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0}\right)}(\varrho ,\tau ;s)={\int }_0^1{\psi }^{\varrho -1}{(1-\psi )}^{\tau -1}\Theta \left({\left\{a_{\iota }\right\}}_{\iota \in {\mathbb{N}}_0};-\frac{s}{\psi (1-\psi )}\right)d\psi ,\end{array}$$

$$\left(\Re \left(s\right)\ge 0;min\{\Re (\varrho ),\Re (\tau \left)\right\}>0\right),$$

where $\Theta \left({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0};z\right)$ indicates the function that has an appropriately bounded sequence ${\left\{a_j\right\}}_{j\in {\mathbb{N}}_0}$ of complex (real) arbitrary that was structured by Srivastava et al. [27] as follows:

$$\Theta \left({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0};z\right)=\left\{\begin{array}{cc}\hfill \sum _{i=0}^{\infty }{a}_j\frac{z^i}{ı!}& \left(\right|z|<R;0<R<\infty ;a_j=1),\\ \\ \hfill {\mathrm{M}}_0{z}^{v}exp\left(z\right)\left[1+\mathcal{O}\left(\frac{1}{z}\right)\right]& \left({\mathrm{M}}_0>0;\Re \left(z\right)\to \infty ;v\in \mathbb{C}\right).\end{array}\right.$$

Diverse investigations studied the types of special functions by Srivastava et al. [28]. More generic incomplete H-functions are constructed alongside their private cases; for example, incomplete Fox–Wright hypergeometric functions. Furthermore, assorted analytical features are acquired for incomplete H-functions that include derivative forms, decomposition and reduction forms, computational representations, and several integral transforms.

The theory of differential subordination of complex-valued functions that was pioneered by Miller and Mocanu [29] generalizes the equation for differential inequality of real-valued functions and is founded on the subordination principle. Its roots may be traced to Lindelöf [30] in 1909. However, Littlewood ([31,32]) and Rogosinski ([33,34]) proposed the phrase and looked at the fundamental consequences of subordination. This concept, which serves as a useful tool, emphasizes the significance of bringing together the presentation of several geometric classes in order to produce some significant results.

Consider the model of open unit complex disk $\mho =\{z\in \mathbb{C}:|z|<1\}$; the term subordinate (superordinate) refers to a mathematical operation, symbolized by ≺, acting on class A of normalized regular functions. It is stated as: the subordinated $\nu$ and $\xi$ in A, referring to $\nu \prec \xi$, if there subsists a function n $\varpi$ in A with $\varpi \left(0\right)=0$ and $\left|\varpi \right(z\left)\right|<1$ achieved that $\nu \left(z\right)=\xi \left(\varpi \right(z\left)\right)$. Particularly, if $\nu$ is univalent in ℧, then $\nu \prec \xi$ if and only if $\nu \left(0\right)=\xi \left(0\right)$ and $\nu (\mho )\subseteq \xi (\mho )$, [30].

Correlated to A, class $ϝ$, which contains all regular meromorphic functions, is constructed as:

$$\begin{array}{c}\hfill \vartheta \left(z\right)=z+{\varkappa }_0+\sum _{\kappa =1}^{\infty }{\varkappa }_{\kappa }{z}^{-\kappa },\end{array}$$

(2)

where z is in the complex exterior disk ${\mho }^{*}=\{z\in \mathbb{C}:1<|z|<\infty \}$. Further, subclass ${ϝ}_0$ involves all functions $\vartheta \in ϝ$ with ${\varkappa }_0\ne 0$ in ${\mho }^{*}$, i.e.

$$\begin{array}{c}\hfill {ϝ}_0=\{\vartheta \in ϝ:{\varkappa }_0\ne 0,z\in {\mho }^{*}\}.\end{array}$$

(3)

Following that, the transformation is stated as follows:

$$\begin{array}{c}\hfill \mathfrak{A}\left(z\right)=\vartheta (1/z)\left(\left|1/z\right|>1\right),\end{array}$$

(4)

then,

$$\begin{array}{c}\hfill \mathfrak{A}\left(z\right)=\frac{1}{z}+\sum _{\kappa =1}^{\infty }{\varkappa }_{\kappa }{z}^{\kappa },\left(0<\left|z\right|<1\right).\end{array}$$

(5)

The term $\Sigma$ of all regular meromorphic functions achieves (5) in the complex punctured disk ${\Delta }^{*}=\{z\in \mathbb{C}:0<|z|<1\}$. One of the captivated cornerstones underlying GFT is the study of the complex (differential and integral) operators, which has recently become fully extensive. The earliest renowned integral operator, Alexander, led to major contributions to GFT, which was settled in 1915 [35]. Then in 1975 and 1984, the famed Ruscheweyh derivative operator [36] and Salagean differential operators [37] first consecutively originated in the work of Ruscheweyh and Salagean; however, of late, they have attracted the discussion of diverse investigators due to their exciting nature. Although originally stimulated by GFT, the vital role of the class of meromorphic functions and the transformation of geometric features between meromorphic function classes is in terms of operators. For instance, studies dealing with operators based on meromorphic functions were conducted in [15,38,39,40,41,42].

According to the term M-LF constructed by (1), the researchers in [43] provided the following new meromorphic function: ${\mathcal{Q}}_{\varrho ,\tau }^{({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0};\sigma )}(z;s)$:

$$\begin{array}{c}\hfill {\mathcal{Q}}_{\varrho ,\tau }^{({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0};\sigma )}(z;s)\end{array}$$

$$\begin{array}{c}\hfill =\frac{B(\sigma ,1-\sigma )\Gamma (\varrho +\tau )}{{\mathcal{B}}^{\left({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0}\right)}(\sigma +1,1-\sigma ;s)}\left[{\mathfrak{M}}_{\varrho ,\tau }^{\left({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0};\sigma \right)}(z;s)-\frac{{\mathcal{B}}^{\left({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0}\right)}(\sigma ,1-\sigma ;s)}{B(\sigma ,1-\sigma )\Gamma \left(\tau \right)}+\frac{{\mathcal{B}}^{\left({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0}\right)}(\sigma +1,1-\sigma ;s)}{B(\sigma ,1-\sigma )\Gamma (\tau +\varrho )}\frac{1}{z}\right]\\ \end{array}$$

$$\begin{array}{c}\hfill =\frac{1}{z}+\sum _{\kappa =1}^{\infty }\frac{\Gamma (\varrho +\tau ){\mathcal{B}}^{\left({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0}\right)}(\sigma +\kappa ,1-\sigma ;s)}{{\mathcal{B}}^{\left({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0}\right)}(\sigma +1,1-\sigma ;s)\Gamma (\varrho \kappa +\tau )}{z}^{\kappa }\\ \end{array}$$

$$\begin{array}{c}\hfill =\frac{1}{z}+\sum _{\kappa =1}^{\infty }{\Lambda }_{\kappa }{z}^{\kappa },\end{array}$$

(6)

where

$${\Lambda }_{\kappa }=\frac{\Gamma (\tau +\varrho ){\mathcal{B}}^{\left({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0}\right)}(\sigma +\kappa ,1-\sigma ;s)}{{\mathcal{B}}^{\left({\left\{a_j\right\}}_{j\in {\mathbb{N}}_0}\right)}(\sigma +1,1-\sigma ;s)\Gamma (\tau +\varrho \kappa )}.$$

Furthermore, the authors in [43] provided the following differential operator constructed by: for $0\le \mu \le 1$ and $\ell \ge 0$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}_{\mu }^0\mathfrak{A}\left(z\right)& =\mathfrak{A}\left(z\right),\hfill \\ \hfill {\mathcal{D}}_{\mu }^1\mathfrak{A}\left(z\right)& ={\mathcal{D}}_{\mu }\mathfrak{A}\left(z\right)={\mathcal{Q}}_{\varrho ,\tau }^{({\left\{a_{\iota }\right\}}_{\iota \in {\mathbb{N}}_0};\sigma )}(z;s)\mathfrak{A}\left(z\right)+\mu \left[z{\left({\mathcal{Q}}_{\varrho ,\tau }^{({\left\{a_{\iota }\right\}}_{\iota \in {\mathbb{N}}_0};\sigma )}(z;s)\right)}^{\prime }\mathfrak{A}\left(z\right)+\frac{1}{z}\right]\hfill \\ & =\frac{1}{z}+\sum _{\kappa =1}^{\infty }(1+\mu \kappa ){\Lambda }_{\kappa }{\varkappa }_{\kappa }{z}^{\kappa },\hfill \\ & ⋮\hfill \\ \hfill {\mathcal{D}}_{\mu }^{\ell }\mathfrak{A}\left(z\right)& ={\mathcal{D}}_{\mu }\left({\mathcal{D}}_{\mu }^{\ell -1}\mathfrak{A}\left(z\right)\right)=\frac{1}{z}+\sum _{\kappa =1}^{\infty }{\left[(1+\mu \kappa ){\Lambda }_{\kappa }\right]}^{\ell }{\varkappa }_{\kappa }{z}^{\kappa }.\hfill \end{array}\end{array}$$

(7)

Next, in [43], for the class $\Omega$ involving regular functions $\wp \left(z\right)$ with $\wp \left(0\right)=1$ and $\mathfrak{A}\in \Sigma$ fulfils an interesting subordination stipulation:

$$\begin{array}{c}\hfill \left(1+\zeta \right)z\left({\mathcal{D}}_{\mu }^{\ell }\mathfrak{A}\left(z\right)\right)+\zeta z^2{\left({\mathcal{D}}_{\mu }^{\ell }\mathfrak{A}\left(z\right)\right)}^{\prime }\prec \wp \left(z\right),\end{array}$$

(8)

then, $\mathfrak{A}$ in ${\Sigma }_{\mu }^{\ell }\left(\zeta ,\wp \right)$, where $\zeta \in \mathbb{C}$ and $\wp \left(z\right)\in \Omega$.

In addition, in [43], in the corresponding operator (7), for $c,m>0$, ${\mathfrak{A}}_j,{\mathfrak{X}}_j\in \Sigma$, $j=1,2,..,$ and ${\eta }_j,{\gamma }_j\ge 0$, $j=1,2,..,m$, we investigated and discussed the following integrodifferential operator: ${\Psi }_m:{\Sigma }^m\to \Sigma$ by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\Psi }_m\left(z\right)=\frac{c}{z^c}{\int }_0^z{\omega }^{c-1}\prod _{j=1}^m{\left(\omega {\mathcal{D}}_{\mu }^{\ell }{\mathfrak{A}}_{ı}\left(\omega \right)\right)}^{{\eta }_j}{\left(-{\omega }^2{\mathcal{D}}_{\mu }^{\ell +1}{\mathfrak{X}}_j\left(\omega \right)\right)}^{{\gamma }_j}d\omega .\end{array}\end{array}$$

(9)

This operator has various specific cases (see [38,43,44,45,46,47,48]).

2. Preliminary Outcomes

The following lemmas are required in order for us to establish our primary findings:

Lemma 1

 ([29]). Let $\mathfrak{J}\left(z\right)$ and $h\left(z\right)$ be two regular functions in Δ. If

$$\begin{array}{c}\hfill \mathfrak{J}\left(z\right)+\frac{1}{c}z{\mathfrak{J}}^{\prime }\left(z\right)\prec h\left(z\right),\end{array}$$

(10)

where $\Re c\ge 0$ and $c\ne 0$, then

$$\begin{array}{c}\hfill \mathfrak{J}\left(z\right)\prec \tilde{h}\left(z\right)=c{z}^{-c}{\int }_0^z{\omega }^{c-1}h\left(\omega \right)d\omega \prec h\left(z\right),\end{array}$$

and $\tilde{h}\left(z\right)$ is the best dominant of (10).

Lemma 2

 ([49]). If $\Re a\ge 0$ and $a\ne 0$, then,

$$\begin{array}{c}\hfill {\Sigma }_{\mu ,\zeta }^{\ell }\left(\tilde{h}\right)\subset {\Sigma }_{\mu ,\zeta }^{\ell }\left(h\right),\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{c}\tilde{h}\left(z\right)=a{z}^{-a}{\int }_0^z{\omega }^{a-1}h\left(\omega \right)d\omega \prec h\left(z\right).\hfill \\ \end{array}\end{array}$$

Lemma 3

 ([49]). If $\mathfrak{A}\left(z\right)\in {\Sigma }_{\mu ,\zeta }^{\ell }\left(h\right),\mathfrak{J}\left(z\right)\in \Sigma$ and

$$\begin{array}{c}\hfill \Re \left(zg\left(z\right)\right)>\frac{1}{2},\left(z\in \Delta \right),\end{array}$$

then,

$$\begin{array}{c}\hfill \left(\mathfrak{A}\ast \mathfrak{J}\right)\left(z\right)\in {\Sigma }_{\mu ,\zeta }^{\ell }\left(h\right).\end{array}$$

3. Main Outcomes

We obtain a few fundamental features in this section, such as the integrodifferential operator property of the extended generalized Mittag-Leffler function in Equation (9) and subordination theory:

Theorem 4.

Let $\mathfrak{A}\left(z\right)\in {\Sigma }_{\mu ,\zeta }^{\ell }\left(h\right)$. Then ${\Psi }_m\left(z\right)$ is the function defined by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\Psi }_m\left(z\right)=\frac{c}{z^c}{\int }_0^z{\omega }^{c-1}\prod _{ı=1}^m{\left(\omega {\mathcal{D}}_{\mu }^{\ell }{\mathfrak{A}}_{ı}\left(\omega \right)\right)}^{{\varrho }_{ı}}{\left(-{\omega }^2{\mathcal{D}}_{\mu }^{\ell +1}{\mathfrak{X}}_{ı}\left(\omega \right)\right)}^{{\varsigma }_{ı}}d\omega ,\end{array}\end{array}$$

(11)

and is in the class ${\Sigma }_{\mu ,\zeta }^{\ell }\left(\tilde{h}\left(z\right)\right)$, where

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\tilde{\Psi }}_m\left(z\right)=c{z}^{1-c}{\int }_0^z{\omega }^{c-2}\prod _{ı=1}^m{\left(\omega {\mathcal{D}}_{\mu }^{\ell }{\mathfrak{A}}_{ı}\left(\omega \right)\right)}^{{\varrho }_{ı}}{\left(-{\omega }^2{\mathcal{D}}_{\mu }^{\ell +1}{\mathfrak{X}}_{ı}\left(\omega \right)\right)}^{{\varsigma }_{ı}}d\omega \end{array}\prec h\left(z\right).\end{array}$$

Proof. 

For $\mathfrak{A}\left(z\right)\in \Sigma$ and $\Re c>1$, we can obtain from (11) that ${\Psi }_m\left(z\right)\in \Sigma$ and

$$\begin{array}{c}\hfill \left(c-1\right)\mathfrak{A}\left(z\right)=c{\Psi }_m\left(z\right)+z{\Psi }_m^{\prime }\left(z\right),{\Psi }_m\left(z\right)\in \Sigma .\end{array}$$

(12)

We define $\Pi \left(z\right)$ by

$$\begin{array}{c}\hfill \Pi \left(z\right)=\left(1+\zeta \right)z\left({\mathcal{D}}_{\mu }^{\ell }{\Psi }_m\left(z\right)\right)+\zeta z^2{\left({\mathcal{D}}_{\mu }^{\ell }{\Psi }_m\left(z\right)\right)}^{\prime }.\end{array}$$

(13)

From (12) and (13), it follows that:

$$\begin{array}{c}\hfill \begin{array}{cc}& \left(1+\zeta \right)z\left({\mathcal{D}}_{\mu }^{\ell }\mathfrak{A}\left(z\right)\right)+\zeta z^2{\left({\mathcal{D}}_{\mu }^{\ell }\mathfrak{A}\left(z\right)\right)}^{\prime }\hfill \\ \\ & =\left(1+\zeta \right)z\left({\mathcal{D}}_{\mu }^{\ell }\left(\frac{c{\Psi }_m\left(z\right)+z{\Psi }_m^{\prime }\left(z\right)}{c-1}\right)\right)+\zeta z^2{\left({\mathcal{D}}_{\mu }^{\ell }\left(\frac{c{\Psi }_m\left(z\right)+z{\Psi }_m^{\prime }\left(z\right)}{c-1}\right)\right)}^{\prime }\hfill \\ \\ & =\frac{c}{c-1}\Pi \left(z\right)+\frac{1}{c-1}\left(z{\Pi }^{\prime }\left(z\right)-\Pi \left(z\right)\right)=\Pi \left(z\right)+\frac{z{\Pi }^{\prime }\left(z\right)}{c-1}.\hfill \\ \end{array}\end{array}$$

(14)

Let $\mathfrak{A}\left(z\right)\in {\Sigma }_{\mu ,\zeta }^{\ell }\left(h\right)$. Then, by (14)

$$\begin{array}{c}\hfill \Pi \left(z\right)+\frac{z{\Pi }^{\prime }\left(z\right)}{c-1}\prec h\left(z\right),\left(\Re c>1\right).\end{array}$$

Hence, we obtain from Lemma 1:

$$\begin{array}{c}\hfill \Pi \left(z\right)\prec \begin{array}{c}\hfill {\tilde{\Psi }}_m\left(z\right)=c{z}^{1-c}{\int }_0^z{\omega }^{c-2}\prod _{ı=1}^m{\left(\omega {\mathcal{D}}_{\mu }^{\ell }{\mathfrak{A}}_{ı}\left(\omega \right)\right)}^{{\varrho }_{ı}}{\left(-{\omega }^2{\mathcal{D}}_{\mu }^{\ell +1}{\mathfrak{X}}_{ı}\left(\omega \right)\right)}^{{\varsigma }_{ı}}d\omega \end{array}\prec h\left(z\right).\end{array}$$

Thus, Lemma 2 contributes to

$$\begin{array}{c}\hfill {\Psi }_m\left(z\right)\in {\Sigma }_{\mu ,\zeta }^{\ell }\left(\tilde{h}\left(z\right)\right)\subset {\Sigma }_{\mu ,\zeta }^{\ell }\left(h\left(z\right)\right).\end{array}$$

□

Theorem 5.

Let ${\Psi }_m\left(z\right)$ be defined in (11) and $\mathfrak{A}\left(z\right)\in \Sigma$. If

$$\begin{array}{c}\hfill \left(1+\alpha \right)z\left({\mathcal{D}}_{\mu }^{\ell }{\Psi }_m\left(z\right)\right)+\alpha z\left({\mathcal{D}}_{\mu }^{\ell }\mathfrak{A}\left(z\right)\right)\prec h\left(z\right),\end{array}$$

(15)

then, ${\Psi }_m\left(z\right)\in {\Sigma }_{\mu }^{\ell }\left(\tilde{h}\left(z\right)\right)={\Sigma }_{\mu ,0}^{\ell }\left(\tilde{h}\left(z\right)\right)$, where $\Re c>1$ and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \tilde{h}\left(z\right)=\frac{\left(c-1\right)}{\alpha }{z}^{\frac{1-c}{\alpha }}{\int }_0^z{\omega }^{\frac{c-1}{\alpha }-1}\prod _{ı=1}^m{\left(\omega {\mathcal{D}}_{\mu }^{\ell }{\mathfrak{A}}_{ı}\left(\omega \right)\right)}^{{\varrho }_{ı}}{\left(-{\omega }^2{\mathcal{D}}_{\mu }^{\ell +1}{\mathfrak{X}}_{ı}\left(\omega \right)\right)}^{{\varsigma }_{ı}}d\omega \end{array}.\end{array}$$

Proof. 

Let us define the regular function $\Pi \left(z\right)$ in $\Delta$ as:

$$\begin{array}{c}\hfill \Pi \left(z\right)=z\left({\mathcal{D}}_{\mu }^{\ell }{\Psi }_m\left(z\right)\right),\end{array}$$

(16)

with $\Pi \left(0\right)=1$, and

$$\begin{array}{c}\hfill z{\Pi }^{\prime }\left(z\right)=\Pi \left(z\right)+z^2{\left({\mathcal{D}}_{\mu }^{\ell }{\Psi }_m\left(z\right)\right)}^{\prime }.\end{array}$$

(17)

By using (12), (15), (16), and (17), we conclude that:

$$\begin{array}{c}\hfill \begin{array}{cc}& \left(1-\alpha \right)z\left({\mathcal{D}}_{\mu }^{\ell }{\Psi }_m\left(z\right)\right)+\alpha z\left({\mathcal{D}}_{\mu }^{\ell }\mathfrak{A}\left(z\right)\right)\hfill \\ & =\left(1-\alpha \right)z\left({\mathcal{D}}_{\mu }^{\ell }{\Psi }_m\left(z\right)\right)+\frac{\alpha }{c-1}\left(cz{\mathcal{D}}_{\mu }^{\ell }{\Psi }_m\left(z\right)\right)+z^2{\left({\mathcal{D}}_{\mu }^{\ell }{\Psi }_m\left(z\right)\right)}^{\prime }\hfill \\ & =\Pi \left(z\right)+\frac{\alpha }{c-1}z{\Pi }^{\prime }\left(z\right)\prec h\left(z\right),\hfill \end{array}\end{array}$$

for $\Re c>1$ and $\alpha >0$.

Therefore, an application of Lemma 1 asserts Theorem 5. □

Theorem 6.

Let $\mathfrak{A}\left(z\right)\in {\Sigma }_{\mu ,\zeta }^{\ell }\left(h\right)$. If ${\Psi }_m\left(z\right)$ is the function given by

$$\begin{array}{c}\hfill {\Psi }_m\left(z\right)=\frac{c-1}{z^c}{\int }_0^z{t}^{c-1}\mathfrak{A}\left(t\right)dt,\left(c>1\right)\end{array}$$

(18)

then,

$$\begin{array}{c}\hfill \sigma \mathfrak{A}\left(\sigma z\right)\in {\Sigma }_{\mu ,\zeta }^{\ell }\left(h\right),\end{array}$$

where

$$\begin{array}{c}\hfill \sigma =\sigma \left(c\right)=\frac{\sqrt{c^2-2\left(c-1\right)}-1}{\left(c-1\right)}\in \left(0,1\right),\end{array}$$

(19)

when

$$\begin{array}{c}\hfill h\left(z\right)=\delta +\left(1-\delta \right)\frac{1+z}{1-z},\left(\delta \ne 1\right).\end{array}$$

(20)

Consequently, the bound σ is sharp.

Proof. 

For ${\Psi }_m\left(z\right)\in {\Sigma }_{\mu ,\zeta }^{\ell }\left(h\right)$, we could verify that: ${\Psi }_m\left(z\right)={\Psi }_m\left(z\right)\ast \frac{z^{-1}}{1-z}$ and $z{\Psi }_m^{\prime }\left(z\right)={\Psi }_m\left(z\right)\ast \left(\frac{1}{{\left(1-z\right)}^2}-\frac{1}{z\left(1-z\right)}\right).$

Then, using (18), we obtain:

$$\begin{array}{c}\hfill \mathfrak{A}\left(z\right)=\frac{c{\Psi }_m\left(z\right)+z{\Psi }_m^{\prime }\left(z\right)}{c-1}=\left(F\ast \mathfrak{J}\right)\left(z\right),\left(z\in {\Delta }^{*},c>1\right),\end{array}$$

(21)

where

$$\begin{array}{c}\hfill \mathfrak{J}\left(z\right)=\frac{1}{c-1}\left(\frac{1}{z{\left(1-z\right)}^2}+\left(c-1\right)\frac{1}{z\left(1-z\right)}\right)\in \Sigma .\end{array}$$

(22)

Now, we prove that:

$$\begin{array}{c}\hfill \Re \left\{z\mathfrak{J}\left(z\right)\right\}>\frac{1}{2},\left(\left|z\right|<\sigma \right),\end{array}$$

(23)

where $\sigma =\sigma \left(c\right)$ is given by (19). Setting

$$\begin{array}{c}\hfill \frac{1}{1-z}={\mathrm{Re}}^{i\theta },\left(R>0,\left|z\right|=r<1\right)\end{array}$$

we have,

$$\begin{array}{c}\hfill cos\theta =\frac{1+R^2\left(1-r^2\right)}{2R}andR\ge \frac{1}{1+r}.\end{array}$$

(24)

By (22) and (24) with $c>1$, we have:

$$\begin{array}{c}\hfill \begin{array}{cc}& 2\Re \left\{z\mathfrak{J}\left(z\right)\right\}=\frac{2}{c-1}\left[\left(c-1\right)Rcos\theta +R^2\left(2{cos}^2\theta -1\right)\right]\hfill \\ & =\frac{1}{c-1}\left[\left(c-1\right)\left(1+R^2\left(1-r^2\right)\right)+{\left(1+R^2\left(1-r^2\right)\right)}^2-R^2\right]\hfill \\ & =\frac{R^2}{c-1}\left[R^2{\left(1-r^2\right)}^2+c\left(1-r^2\right)-1\right]+1\ge \frac{R^2}{c-1}\left[{\left(1-r^2\right)}^2+c\left(1-r^2\right)-1\right]+1\hfill \\ & =\frac{R^2}{c-1}\left[\left(1-c\right)r^2+c-2r\right]+1.\hfill \end{array}\end{array}$$

This would eventually give (23); hence,

$$\begin{array}{c}\hfill \Re \left\{z\sigma \mathfrak{J}\left(\sigma z\right)\right\}>\frac{1}{2},\left(z\in \Delta \right).\end{array}$$

(25)

Let ${\Psi }_m\left(z\right)\in {\Sigma }_{\mu ,\zeta }^{\ell }\left(h\right)$. Using (21) and (25) with Lemma 3, we have:

$$\begin{array}{c}\hfill \sigma \mathfrak{A}\left(\sigma z\right)={\Psi }_m\left(z\right)\ast \sigma \mathfrak{J}\left(\sigma z\right)\in {\Sigma }_{\mu ,\zeta }^{\ell }\left(h\right).\end{array}$$

For $h\left(z\right)$ defined by (20), function ${\Psi }_m\left(z\right)\in \Sigma$ is given by:

$$\begin{array}{c}\hfill \left(1+\zeta \right)z\left({\mathcal{D}}_{\mu }^{\ell }{\Psi }_m\left(z\right)\right)+\zeta z^2{\left({\mathcal{D}}_{\mu }^{\ell }{\Psi }_m\left(z\right)\right)}^{\prime }=\delta +\left(1-\delta \right)\frac{1+z}{1-z},\left(\delta \ne 1\right).\end{array}$$

(26)

By using (13), (14) and (26), we obtain the following:

$$\begin{array}{c}\hfill \begin{array}{cc}& \left(1+\zeta \right)z\left({\mathcal{D}}_{\mu }^{\ell }\mathfrak{A}\left(z\right)\right)+\zeta z^2{\left({\mathcal{D}}_{\mu }^{\ell }\mathfrak{A}\left(z\right)\right)}^{\prime }\hfill \\ & =\delta +\left(1-\delta \right)\frac{1+z}{1-z}+\frac{z}{c-1}{\left(\delta +\left(1-\delta \right)\frac{1+z}{1-z}\right)}^{\prime }\hfill \\ & =\delta +\frac{\left(1-\delta \right)\left(c+2z-1+\left(1-c\right)z^2\right)}{\left(c-1\right){\left(1-z\right)}^2}=\delta ,\left(\sigma =-z\right).\hfill \end{array}\end{array}$$

Hence, for each $c\left(c>1\right)$, the bound $\sigma =\sigma \left(c\right)$ could not be increased. □

4. Conclusions

Many researchers have studied the theory of analytical functions. The investigation of inequalities and special functions in complex analysis is a more focused topic. Numerous studies based on different kinds of analytical functions are cited in the literature. This expansion is closely related to the relationship between geometric behavior and analytical structure. Motivated by this technique, in this analytical inquiry, utilizing the extended generalized M-LF and subordination theory, a new complex integrodifferential operator that is linked to both the Mittag-Leffler function and the meromorphic functions in the punctured unit disk was constructed and successfully implemented. By utilizing this new operator in this article, we covered several fascinating advantages of new geometric meromorphic analytic function subclasses.