New Subclass of Meromorphic Functions Defined via Mittag–Leffler Function on Hilbert Space

Abstract

In this paper, a novel class of meromorphic functions associated with the Mittag–Leffler function $E_{\mu ,\vartheta }\left(z\right)$ is introduced using the Hilbert space operator. In the punctured symmetric domain ${⋓}^{\ast }$, essential properties of this class are systematically investigated. These properties include coefficient inequalities, growth and distortion bounds, as well as weighted and arithmetic mean estimates. Furthermore, the extreme points and radii of geometric properties such as close-to-convexity, starlikeness, and convexity are analyzed in detail. Additionally, the Hadamard product (or convolution) is explored to demonstrate the algebraic structure and stability of the introduced function class under this operation. Integral mean inequalities are also established to provide further insights into the behavior of these functions within the given domain.

1. Introduction

The study of meromorphic functions has gained considerable attention in complex analysis due to their rich mathematical structure and diverse applications in both theoretical and applied sciences. These functions play a vital role in operator theory, functional equations, and geometric function theory, and they often appear in modeling complex phenomena in physics and engineering.

It is worth noting that recent contributions such as in [1] have focused on defining meromorphic function classes via specific integral operators on Hilbert spaces. On the other hand, more recent works (see [2,3]) have highlighted numerical tools and theoretical frameworks such as the Uncertainty Principle in radial basis functions (RBFs) and the concept of the effective condition number, drawing meaningful links between functional analysis and numerical modeling. These developments motivate a deeper investigation into the interplay between operator norms, shape parameters, and the geometric structure of meromorphic function spaces.

In the punctured symmetric domain ${⋓}^{\ast }=\{z\in \mathbb{C}:0<|z|<1\}=⋓\setminus \left\{0\right\},$ let $\mathsf{\Sigma }$ represent the class of meromorphic functions of the form

$$\mathfrak{J}\left(z\right)={\displaystyle \frac{1}{z}}+\sum _{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n}}{z}^{\mathfrak{n}},$$

(1)

which are analytic in symmetric domain ${⋓}^{\ast }$.

Let ${\mathsf{\Sigma }}^{\ast }(\wp )$ and ${\mathsf{\Sigma }}_k(\wp )$ represent subclasses of $\mathsf{\Sigma }$ consisting of meromorphically starlike functions of order ℘ and meromorphically convex functions of order ℘, respectively; see [4,5,6,7]. A function $\mathfrak{J}$ of the form (1) pertains to the class ${\mathsf{\Sigma }}^{\ast }(\wp )$ if it fulfills the subsequent requirement:

$$\Re \left\{-{\displaystyle \frac{z{\mathfrak{J}}^{\prime }\left(z\right)}{\mathfrak{J}\left(z\right)}}\right\}>\wp (z\in {⋓}^{\ast }),$$

and $\mathfrak{J}\in {\mathsf{\Sigma }}_k(\wp )$ if it satisfies

$$\Re \left\{-\left(1+{\displaystyle \frac{z{\mathfrak{J}}^{″}\left(z\right)}{{\mathfrak{J}}^{\prime }\left(z\right)}}\right)\right\}>\wp (z\in {⋓}^{\ast }).$$

Let $\mathfrak{J}\ast \mathfrak{K}$ denote the Hadamard product (or convolution) of the functions $\mathfrak{J}$ and $\mathfrak{K}$, that is, if $\mathfrak{J}$ is given by Equation (1) and the function $\mathfrak{K}\left(z\right)$ is given by

$$\mathfrak{K}\left(z\right)={\displaystyle \frac{1}{z}}+\sum _{\mathfrak{n}=1}^{\infty }{b}_{\mathfrak{n}}{z}^{\mathfrak{n}},$$

(2)

then

$$(\mathfrak{J}\ast \mathfrak{K})\left(z\right):={\displaystyle \frac{1}{z}}+\sum _{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n}}{b}_{\mathfrak{n}}{z}^{\mathfrak{n}}.$$

(3)

Several subclasses of the class $\mathsf{\Sigma }$ have been introduced and studied by various researchers. In recent years, a certain subfamily of meromorphic functions associated with multiple families of integral and differential operators has been defined and extensively investigated (see, for example [8,9,10,11]).

Next, we introduce the well-known Mittag–Leffler function $E_{\mu }\left(z\right)$, originally defined by Mittag–Leffler [12], along with its generalized form $E_{\mu ,\vartheta }\left(z\right)$, which was later introduced by Wiman (see [13,14]). These functions are given by the following definitions:

$$E_{\mu }\left(z\right)=\sum _{\mathfrak{n}=0}^{\infty }{\displaystyle \frac{z^{\mathfrak{n}}}{\mathsf{\Gamma }(\mu \mathfrak{n}+1)}}=E_{\mu ,1}\left(z\right),$$

(4)

and

$$E_{\mu ,\vartheta }\left(z\right)=\sum _{\mathfrak{n}=0}^{\infty }{\displaystyle \frac{z^{\mathfrak{n}}}{\mathsf{\Gamma }(\mu \mathfrak{n}+\vartheta )}}.$$

(5)

where $z\in \mathbb{C};min[\Re (\vartheta ),\Re (\mu \left)\right]>0$.

The Mittag–Leffler functions appear in various applied sciences and mathematical models due to their ability to generalize the exponential function. These functions may be found in many different domains, such as complex system studies, random walks, Lévy flights, super-diffusive transportation issues, and solutions to fractional differential equations. Their role is particularly prominent in capturing memory effects and anomalous diffusion behavior in physical and engineering systems.

Various characteristics of the Mittag–Leffler function $E_{\mu }\left(z\right)$ and $E_{\mu ,\vartheta }\left(z\right)$, along with their generalizations, are extensively studied in the modern literature and are central to many recent investigations (see [4,12,13,15]).

We note that the function given by (5) is not part of class $\mathsf{\Sigma }$. Therefore, the function ${\mathsf{\Xi }}_{\mu ,\vartheta }\left(z\right)$ is then normalized on the basis of the following:

$${\mathsf{\Xi }}_{\mu ,\vartheta }\left(z\right)=z^{-1}\mathsf{\Gamma }\left(\vartheta \right)E_{\mu ,\vartheta }\left(z\right)=z^{-1}+\sum _{\mathfrak{n}=1}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\vartheta \right)z^{\mathfrak{n}}}{\mathsf{\Gamma }(\mu \mathfrak{n}+\vartheta +\mu )}},$$

(6)

where $z\in \mathbb{C};min[\Re (\vartheta ),\Re (\mu \left)\right]>0$ and $\vartheta \ne 0,-1,-2,\dots$

Using the function ${\mathsf{\Xi }}_{\mu ,\vartheta }\left(z\right)$ given in (6), we define a new operator ${\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}\left(z\right)\right]:\mathsf{\Sigma }\to \mathsf{\Sigma }$ through the Hadamard product as follows [11,13,15]:

$${\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}\left(z\right)\right]=\mathfrak{J}\left(z\right)\ast {\mathsf{\Xi }}_{\mu ,\vartheta }\left(z\right)=z^{-1}+\sum _{\mathfrak{n}=1}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\vartheta \right)}{\mathsf{\Gamma }(\mu \mathfrak{n}+\vartheta +\mu )}}{a}_{\mathfrak{n}}{z}^{\mathfrak{n}}=z^{-1}+\sum _{\mathfrak{n}=1}^{\infty }\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}{z}^{\mathfrak{n}},$$

(7)

where $\psi (\mathfrak{n},\mu ,\vartheta )={\displaystyle \frac{\mathsf{\Gamma }\left(\vartheta \right)}{\mathsf{\Gamma }(\mu \mathfrak{n}+\vartheta +\mu )}}$.

Now, we will present some key properties of Hilbert spaces and their connection to Euclidean geometry. We will begin by introducing the main concepts such as the inner product, norm, and Hilbert spaces.

Inner product spaces extend Euclidean geometry to more abstract vector spaces, including infinite-dimensional ones. For a vector space V over the field $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, the inner product is a mapping $\langle ·,·\rangle :V\times V\to \mathbb{F}$ satisfying the following:

• Additivity: $\langle x+y,z\rangle =\langle x,z\rangle +\langle y,z\rangle$;
• Homogeneity: $\langle ax,y\rangle =a\langle x,y\rangle$;
• Conjugate Symmetry: $\langle x,y\rangle =\overline{\langle y,x\rangle }$;
• Positive Definite: $\langle x,x\rangle >0$ if $x\ne 0$.

In the above, ${(.)}^{\ast }$ denotes the complex conjugate. The first three axioms imply the following additional properties:

• $\langle x,y+z\rangle =\langle x,y\rangle +\langle x,z\rangle$;
• $\langle x,ay\rangle =a^{\ast }\langle x,y\rangle$.

The norm in an inner product space is defined by $\parallel x\parallel =\sqrt{\langle x,x\rangle }$, where the inner product induces the geometric structure. A Hilbert space is one that is complete with respect to this norm. For more details, see [16].

Lemma 1

(Cauchy–Schwarz Inequality). For any $x,y\in V$,

$$|\langle x,y\rangle |\le \parallel x\parallel ·\parallel y\parallel .$$

Equality holds if and only if x and y are linearly dependent; for further details, refer to [17].

Orthogonality: Two vectors x and y are orthogonal, denoted by $x\perp y$, if $\langle x,y\rangle =0$.

Theorem 1

(Generalized Pythagorean Theorem). If $x\perp y$, then

$${\parallel x+y\parallel }^2={\parallel x\parallel }^2+{\parallel y\parallel }^2.$$

The converse holds in real spaces; for further details, refer to [18].

Let $\mathcal{H}$ be a Hilbert space over the complex field, and let $L\left(\mathcal{H}\right)$ denote the algebra of all bounded linear operators on $\mathcal{H}$. Suppose that $\mathfrak{J}$ is a complex function that is analytic within a region $\mathsf{\Omega }$ of the complex plane, where $\mathsf{\Omega }$ contains the spectrum $\sigma (\top )$ of a bounded linear operator ⊤. The operator $\mathfrak{J}(\top )$ on $\mathcal{H}$ is then defined via the Riesz–Dunford integral (see [1,5]).

$$\mathfrak{J}(\top )={\displaystyle \frac{1}{2\pi i}}{\int }_C{(zI-\top )}^{-1}\mathfrak{J}\left(z\right)dz,$$

Here, I denotes the identity operator on $\mathcal{H}$, and C is a positively oriented, simple, closed, rectifiable contour that encircles the spectrum $\sigma (\top )$ in its interior. Moreover, the operator $\mathfrak{J}(\top )$ can also be represented using the following series expansion:

$$\mathfrak{J}(\top )=\sum _{\mathfrak{n}=0}^{\infty }{\displaystyle \frac{{\mathfrak{J}}^{\left(\mathfrak{n}\right)}\left(0\right)}{\mathfrak{n}!}}{\top }^{\mathfrak{n}},$$

which converges in the norm topology (for more details, see [1,19,20]). The set of all functions $\mathfrak{J}\in \mathsf{\Sigma }$ satisfying $a_{\mathfrak{n}}\ge 0$ is represented by ${\mathsf{\Sigma }}_p$.

Definition 1.

For $0\le ⋋<1$ and $0\le \varsigma <1$, a function $\mathfrak{J}\in {\mathsf{\Sigma }}_p$ given by Equation (1) belongs to the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$ if

$$\Re \left({\displaystyle \frac{{\top }^3{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{‴}}{⋋{\top }^2{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{″}+2(3+⋋)\top {\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{\prime }}}\right)>\varsigma ,$$

(8)

where ${\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]$ is given by Equation (7).

Remark 1.

Consider the generalized subclass of meromorphic functions $\mathfrak{A}(\varsigma ,⋋)$. For $0\le \alpha <1$ and $0\le ⋋<1$, a function $\mathfrak{J}\in {\mathsf{\Sigma }}_p$, described by Equation (1), belongs to this class if and only if

$$\Re \left({\displaystyle \frac{z^3{\mathfrak{J}}^{‴}\left(z\right)}{⋋z^2{\mathfrak{J}}^{″}\left(z\right)+2(3+⋋)z{\mathfrak{J}}^{\prime }\left(z\right)}}\right)>\alpha .$$

(9)

Lemma 2

([1]). Let $\varpi =u+iv$. Then, $R\left(\varpi \right)>\alpha \iff |\varpi -1|<|\varpi +1-2\alpha |.$

By applying Lemma 2, we obtain an equivalent Definition 1.

Definition 2.

For $0\le ⋋<1$ and $0\le \varsigma <1$, a function $\mathfrak{J}\in {\mathsf{\Sigma }}_p$ given by (1) is in the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$ if the following inequality is satisfied:

$$\Vert {\top }^3{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{‴}-\left(⋋{\top }^2{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{″}+2(3+⋋)\top {\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{\prime }\right)\Vert$$

$$<\Vert {\top }^3{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{‴}+(1-2\varsigma )\left(⋋{\top }^2{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{″}+2(3+⋋)\top {\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{\prime }\right)\Vert$$

for all operators ⊤ with $\parallel \top \parallel <1$ and $\top \ne \mathsf{\Theta }$ (where Θ is the zero operator on H).

In the next sections, we analyze certain geometric properties after establishing coefficient bounds, including growth and distortion estimates, along with arithmetic and weighted means. We also explore geometric features such as close-to-convexity, starlikeness, and convexity, determining their respective radii. Moreover, we examine the behavior of the Hadamard product (or convolution) within Hilbert spaces.

2. Cofficient Bounds

We begin by describing the properties of the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$ by determining the necessary and sufficient conditions for a function to belong to it. This characterization naturally leads to coefficient bounds.

Theorem 2.

A function $\mathfrak{J}\in {\mathsf{\Sigma }}_p$ given by Equation (1) belongs to the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$ for all proper contractions T with $\top \ne \mathsf{\Theta }$ if and only if

$$\sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}\le 12(\varsigma -1),$$

(10)

the obtained result is optimal for the function

$$\mathfrak{J}(\top )={\displaystyle \frac{1}{\top }}+\sum _{\mathfrak{n}=1}^{\infty }{\displaystyle \frac{12(\varsigma -1)}{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}}{\top }^{\mathfrak{n}},(\mathfrak{n}\ge 1).$$

(11)

Proof. 

Assuming that (10) holds, and applying the norm to Lemma 2 in accordance with Definition 1, we obtain the following inequalities:

$$\Vert {\top }^3{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{‴}-\left(⋋{\top }^2{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{″}+2(3+⋋)\top {\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{\prime }\right)\Vert$$

$$-\Vert {\top }^3{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{‴}+(1-2\varsigma )\left(⋋{\top }^2{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{″}+2(3+⋋)\top {\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{\prime }\right)\Vert <0.$$

(12)

By substituting the values of the derivatives into the function ${\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]$, we obtain the following result:

$$\Vert \sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}({\mathfrak{n}}^2-3\mathfrak{n}-⋋\mathfrak{n}-⋋-4)\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}{\top }^{\mathfrak{n}}\Vert$$

$$-\Vert \sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[{\mathfrak{n}}^2+\mathfrak{n}(⋋-2\varsigma ⋋-3)-2(⋋\varsigma +2)+⋋]\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}{\top }^{\mathfrak{n}}+12(\varsigma -1){\top }^{-1}\Vert .$$

Then, by applying the triangle inequality, we obtain the following result:

$$\le \sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}({\mathfrak{n}}^2-3\mathfrak{n}-⋋\mathfrak{n}-⋋-4)\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}{\Vert \top \Vert }^{\mathfrak{n}}-12(\varsigma -1)\Vert {\top }^{-1}\Vert$$

$$+\sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[{\mathfrak{n}}^2+\mathfrak{n}(⋋-2\varsigma ⋋-3)-2(⋋\varsigma +2)+⋋]\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}\Vert {\top }^{\mathfrak{n}}\Vert .$$

So,

$$\sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}{\Vert \top \Vert }^{\mathfrak{n}}-12(\varsigma -1)\Vert {\top }^{-1}\Vert ,$$

$$\le 12(\varsigma -1)-12(\varsigma -1)=0.$$

Thus, $\mathfrak{J}\in {\mathsf{\Sigma }}_p$ is in the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. Conversely, suppose that $\mathfrak{J}\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$, that is,

$$\Vert {\top }^3{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{‴}-\left(⋋{\top }^2{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{″}+2(3+⋋)\top {\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{\prime }\right)\Vert$$

$$<\Vert {\top }^3{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{‴}+(1-2\varsigma )\left(⋋{\top }^2{\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{″}+2(3+⋋)\top {\left({\mathfrak{B}}_{\mathsf{\Sigma },\vartheta }^{\mathfrak{n},\mu }\left[\mathfrak{J}(\top )\right]\right)}^{\prime }\right)\Vert .$$

From the last inequality, the following is obtained:

$$=\Vert \sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}({\mathfrak{n}}^2-3\mathfrak{n}-⋋\mathfrak{n}-⋋-4)\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}{\top }^{\mathfrak{n}+1}\Vert$$

$$<\Vert \sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[{\mathfrak{n}}^2+\mathfrak{n}(⋋-2\varsigma ⋋-3)-2(⋋\varsigma +2)+⋋]\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}{\top }^{\mathfrak{n}+1}+12(\varsigma -1)\Vert$$

Selecting $\top =rI$ ($0<r<1$) in the above inequality, it is found that

$${\displaystyle \frac{{\sum }_{\mathfrak{n}=1}^{\infty }\mathfrak{n}({\mathfrak{n}}^2-3\mathfrak{n}-⋋\mathfrak{n}-⋋-4)\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}{r}^{\mathfrak{n}+1}}{12(\varsigma -1)+{\sum }_{\mathfrak{n}=1}^{\infty }\mathfrak{n}[{\mathfrak{n}}^2+\mathfrak{n}(⋋-2\varsigma ⋋-3)-2(⋋\varsigma +2)+⋋]\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}{r}^{\mathfrak{n}+1}}}<1.$$

As $r\to 1^{-}$, (10) is obtained. □

Corollary 1.

If the function $\mathfrak{J}\in {\mathsf{\Sigma }}_p$, defined by (1), belongs to the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$, then

$$\sum _{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n}}\le {\displaystyle \frac{12(\varsigma -1)}{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}},(\mathfrak{n}\ge 1).$$

(13)

3. Closure Properties of Arithmetic Mean and Weighted Mean

This section focuses on studying the closure of the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$ in terms of the arithmetic mean and weighted mean.

Definition 3.

The weighted mean ${\mathbb{M}}_{\mathcal{Q}}(\top )$ of $\mathfrak{J}(\top )$ and $\mathfrak{K}(\top )$ is defined by

$${\mathbb{M}}_{\mathcal{Q}}(\top )={\displaystyle \frac{1}{2}}\left[(1-\mathcal{Q})\mathfrak{J}(\top )+(1+\mathcal{Q})\mathfrak{K}(\top )\right],0<\mathcal{Q}<1.$$

The following theorem establishes the connection between the weighted mean and the specified class.

Theorem 3.

Let $\mathfrak{J}(\top )$ and $\mathfrak{K}(\top )$ be in the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. Then, the weighted mean of $\mathfrak{J}(\top )$ and $\mathfrak{K}(\top )$ is present in the class as well, i.e., ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$.

Proof. 

By the definition of the weighted mean, the following is obtained:

$${\mathbb{M}}_{\mathcal{Q}}(\top )={\displaystyle \frac{1}{2}}\left[(1-\mathcal{Q})\mathfrak{J}(\top )+(1+\mathcal{Q})\mathfrak{K}(\top )\right]$$

(14)

$$={\displaystyle \frac{1}{2}}\left((1-\mathcal{Q})\left({\displaystyle \frac{1}{\top }}+\sum _{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n}}{\top }^{\mathfrak{n}}\right)+(1+\mathcal{Q})\left({\displaystyle \frac{1}{\top }}+\sum _{\mathfrak{n}=1}^{\infty }{b}_{\mathfrak{n}}{\top }^{\mathfrak{n}}\right)\right)$$

$$={\displaystyle \frac{1}{\top }}+\sum _{\mathfrak{n}=1}^{\infty }{\displaystyle \frac{1}{2}}[(1-\mathcal{Q})a_{\mathfrak{n}}+(1+\mathcal{Q})b_{\mathfrak{n}}]{\top }^{\mathfrak{n}}.$$

Since $\mathfrak{J}(\top )$ and $\mathfrak{K}(\top )$ belong to the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$, based on Theorem 2, the following is implied:

$$\sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}\le 12(\varsigma -1),$$

(15)

and

$$\sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )b_{\mathfrak{n}}\le 12(\varsigma -1).$$

(16)

Hence,

$$\sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )\left({\displaystyle \frac{1}{2}}[(1-\mathcal{Q})a_{\mathfrak{n}}+(1+\mathcal{Q})b_{\mathfrak{n}}]\right)$$

$$={\displaystyle \frac{1}{2}}[(1-\mathcal{Q})\sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}$$

$$+{\displaystyle \frac{1}{2}}(1+\mathcal{Q})\sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )b_{\mathfrak{n}}],$$

$$\le {\displaystyle \frac{12(1-\mathcal{Q})(\varsigma -1)}{2}}+{\displaystyle \frac{12(1+\mathcal{Q})(\varsigma -1)}{2}}=12(\varsigma -1).$$

□

Theorem 4.

Let ${\mathfrak{J}}_1(\top ),{\mathfrak{J}}_2(\top ),\dots ,{\mathfrak{J}}_{\iota }(\top )$, which is defined by

$${\mathfrak{J}}_{\tau }(\top )={\displaystyle \frac{1}{\top }}+\sum _{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n},\tau }{\top }^{\mathfrak{n}},a_{\mathfrak{n},\tau }\ge 0,\tau =1,2,\dots ,\iota ,\mathfrak{n}\ge 2,$$

(17)

be members of the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. Then, the arithmetic mean of ${\mathfrak{J}}_{\tau }(\top )$, where $\tau =1,2,\dots ,\iota$ and is defined by

$$\mathcal{M}(\top )={\displaystyle \frac{1}{\iota }}\sum _{\tau =1}^{\iota }{\mathfrak{J}}_{\tau }(\top ),$$

is also in the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$.

Proof. 

By the definitions of ${\mathfrak{J}}_{\tau }(\top )$ and $\mathcal{M}(\top )$, we have

$$\mathcal{M}(\top )={\displaystyle \frac{1}{\iota }}\sum _{\tau =1}^{\iota }\left({\displaystyle \frac{1}{\top }}+\sum _{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n},\tau }{\top }^{\mathfrak{n}}\right)={\displaystyle \frac{1}{\top }}+\sum _{\mathfrak{n}=1}^{\infty }\left(\sum _{\tau =1}^{\iota }{\displaystyle \frac{1}{\iota }}{a}_{\mathfrak{n},\tau }\right){\top }^{\mathfrak{n}}.$$

(18)

Since ${\mathfrak{J}}_{\tau }(\top )\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$ for every $\tau =1,2,\dots ,\iota$, by Theorem 2, it is determined that

$$\sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )\left(\sum _{\tau =1}^{\iota }{\displaystyle \frac{1}{\iota }}{a}_{\mathfrak{n},\tau }\right),$$

(19)

$$={\displaystyle \frac{1}{\iota }}\sum _{\tau =1}^{\iota }\left(\sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n},\tau }\right),$$

$$\le {\displaystyle \frac{1}{\iota }}\sum _{\tau =1}^{\iota }12(\varsigma -1)=12(\varsigma -1).$$

(20)

The proof is complete. □

4. Growth and Distortion Bounds

This section focuses on deriving growth and distortion estimates for the family ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$.

Theorem 5.

If $\mathfrak{J}\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$, then for $0<\left|z\right|=r<1$, we have

$${\displaystyle \frac{1}{\Vert \top \Vert }}-\Vert \top \Vert {\displaystyle \frac{12(1-\varsigma )}{[12+⋋]\psi (1,\mu ,\vartheta )}}\le \Vert \mathfrak{J}(\top )\Vert \le {\displaystyle \frac{1}{\Vert \top \Vert }}+\Vert \top \Vert {\displaystyle \frac{12(1-\varsigma )}{[12+⋋]\psi (1,\mu ,\vartheta )}};$$

(21)

this result attains sharpness for

$$\mathfrak{J}\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{12(1-\varsigma )}{[12+⋋]\psi (1,\mu ,\vartheta )}}z.$$

(22)

Proof. 

Suppose $\mathfrak{J}\left(z\right)$ is in the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. By Theorem 2, we have

$$\sum _{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n}}\le {\displaystyle \frac{12(1-\varsigma )}{[12+⋋]\psi (1,\mu ,\vartheta )}}.$$

(23)

Also, if $\mathfrak{J}(\top )={\top }^{-1}+{\sum }_{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n}}{\top }^{\mathfrak{n}}$, then

$${\displaystyle \frac{1}{\Vert \top \Vert }}-\sum _{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n}}{\Vert \top \Vert }^{\mathfrak{n}}\le \Vert \mathfrak{J}(\top )\Vert \le {\displaystyle \frac{1}{\Vert \top \Vert }}+\sum _{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n}}{\Vert \top \Vert }^{\mathfrak{n}}.$$

(24)

Since $\Vert \top \Vert <1$, the above inequality becomes

$${\displaystyle \frac{1}{\Vert \top \Vert }}-\Vert \top \Vert \sum _{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n}}\le \Vert \mathfrak{J}(\top )\Vert \le {\displaystyle \frac{1}{\Vert \top \Vert }}+\Vert \top \Vert \sum _{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n}}.$$

(25)

Using (23), we obtain the result. □

Theorem 6.

If $\mathfrak{J}\left(z\right)\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$, then

$${\displaystyle \frac{1}{{\Vert \top \Vert }^2}}-{\displaystyle \frac{12(1-\varsigma )}{[12+⋋]\psi (1,\mu ,\vartheta )}}\le \Vert {\mathfrak{J}}^{\prime }(\top )\Vert \le {\displaystyle \frac{1}{{\Vert \top \Vert }^2}}+{\displaystyle \frac{12(1-\varsigma )}{[12+⋋]\psi (1,\mu ,\vartheta )}}.$$

(26)

The outcome is thus pronounced for

$$\mathfrak{J}\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{12(1-\varsigma )}{[12+⋋]\psi (1,\mu ,\vartheta )}}z.$$

(27)

5. Convex Set

Theorem 7.

The class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$ is closed under convex combinations.

Proof. 

Let

$$\mathfrak{J}(\top )={\displaystyle \frac{1}{\top }}+\sum _{\mathfrak{n}=1}^{\infty }{a}_{\mathfrak{n}}{\top }^{\mathfrak{n}},\mathrm{and}\mathfrak{K}(\top )={\displaystyle \frac{1}{\top }}+\sum _{\mathfrak{n}=1}^{\infty }{b}_{\mathfrak{n}}{\top }^{\mathfrak{n}}$$

(28)

be in the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$, satisfying (15) and (16).

Also, the function $\hslash (\top )$ is defined for $0\le \varrho \le 1$, as follows:

$$\hslash (\top )=\varrho \mathfrak{J}(\top )+(1-\varrho )\mathfrak{K}(\top ).$$

By substitution, the following is obtained:

$$\hslash (\top )={\displaystyle \frac{1}{\top }}+\sum _{\mathfrak{n}=1}^{\infty }\left[\varrho a_{\mathfrak{n}}+(1-\varrho )b_{\mathfrak{n}}\right]{\top }^{\mathfrak{n}}.$$

Now, the calculation proceeds as follows:

$$\hslash (\top )=\varrho \sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )a_{\mathfrak{n}}$$

$$+(1-\varrho )\sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )b_{\mathfrak{n}},$$

(29)

$$\le 12\varrho (\varsigma -1)+12(1-\varrho )(\varsigma -1)=12(\varsigma -1).$$

Thus, $\hslash (\top )\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. □

6. Extreme Points

Theorem 8.

If

$${\mathfrak{J}}_0(\top )={\displaystyle \frac{1}{\top }}and{\mathfrak{J}}_{\mathfrak{n}}(\top )={\displaystyle \frac{1}{\top }}+{\displaystyle \frac{12(\varsigma -1)}{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}}{\top }^{\mathfrak{n}},$$

(30)

then $\mathfrak{J}\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$ if and only if it can be represented in the form

$$\mathfrak{J}(\top )=\sum _{\mathfrak{n}=0}^{\infty }{\ell }_{\mathfrak{n}}{\mathfrak{J}}_{\mathfrak{n}}(\top ),{\ell }_{\mathfrak{n}}\ge 0,\sum _{\mathfrak{n}=0}^{\infty }{\ell }_{\mathfrak{n}}=1.$$

(31)

where $\mathfrak{n}=1,2,\dots$

Proof. 

Assume that $\mathfrak{J}(\top )={\sum }_{\mathfrak{n}=0}^{\infty }{\ell }_{\mathfrak{n}}{\mathfrak{J}}_{\mathfrak{n}}(\top ),({\ell }_{\mathfrak{n}}\ge 0,\mathfrak{n}=0,1,2,\dots ;{\sum }_{\mathfrak{n}=0}^{\infty }{\ell }_{\mathfrak{n}}=1).$ Then, it is obtained that

$$\mathfrak{J}(\top )=\sum _{\mathfrak{n}=0}^{\infty }{\ell }_{\mathfrak{n}}{\mathfrak{J}}_{\mathfrak{n}}(\top )={\ell }_0{\mathfrak{J}}_0(\top )+\sum _{\mathfrak{n}=1}^{\infty }{\ell }_{\mathfrak{n}}{\mathfrak{J}}_{\mathfrak{n}}(\top )$$

(32)

$$={\displaystyle \frac{1}{\top }}+\sum _{\mathfrak{n}=1}^{\infty }{\ell }_{\mathfrak{n}}{\displaystyle \frac{12(\varsigma -1)}{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}}{\top }^{\mathfrak{n}}.$$

(33)

Therefore,

$${\sum }_{\mathfrak{n}=1}^{\infty }\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta ){\ell }_{\mathfrak{n}}{\displaystyle \frac{6(\varsigma -1)}{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}}{\top }^{\mathfrak{n}}$$

(34)

$$=12(\varsigma -1)\sum _{\mathfrak{n}=1}^{\infty }{\ell }_{\mathfrak{n}}$$

$$=12(\varsigma -1)(1-{\ell }_0)\le 12(\varsigma -1).$$

(35)

Conversely, suppose that $\mathfrak{J}\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. Since, by Corollary 1,

$$a_{\mathfrak{n}}\le {\displaystyle \frac{12(\varsigma -1)}{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}}(\mathfrak{n}\ge 1),$$

(36)

setting

$${\ell }_{\mathfrak{n}}={\displaystyle \frac{12(\varsigma -1)}{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}}{a}_{\mathfrak{n}}(\mathfrak{n}\ge 1)$$

(37)

and ${\ell }_0=1-{\sum }_{\mathfrak{n}=1}^{\infty }{\ell }_{\mathfrak{n}},$ we obtain $\mathfrak{J}(\top )={\ell }_0{\mathfrak{J}}_0(\top )+{\sum }_{\mathfrak{n}=1}^{\infty }{\ell }_{\mathfrak{n}}{\mathfrak{J}}_{\mathfrak{n}}(\top )$. The proof is complete. □

7. Radii of Starlikeness and Convexity for the Class ${\mathfrak{A}}_{\mathbf{\vartheta }}^{\mathbf{\mu }}(\mathbf{\varsigma },⋋,\top )$

In this section, we determine the radii of meromorphic close-to-convexity, starlikeness, and convexity for functions $\mathfrak{J}$ belonging to the family ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$.

Theorem 9.

Let $\mathfrak{J}\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. Then, $\mathfrak{J}$ is meromorphically close-to-convex of order ℘ (where $0\le \wp <1$) in the disk $\left|z\right|<r_1$, where

$$r_1=\underset{\mathfrak{n}\ge 1}{inf}{\left({\displaystyle \frac{(1-\wp )\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12\mathfrak{n}(\varsigma -1)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{n}+1}}}.$$

The result is precise for the extremal function $\mathfrak{J}$ defined by Equation (1).

Proof. 

This is enough to demonstrate

$$\Vert {\displaystyle \frac{{\mathfrak{J}}^{\prime }(\top )}{{\top }^{-2}}}+1\Vert <1-\wp .$$

According to Theorem 2, the following inequality is established:

$$\sum _{\mathfrak{n}=1}^{\infty }{\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]{\mathsf{\Psi }}_{\mathfrak{n}}}{12(\varsigma -1)}}{a}_{\mathfrak{n}}\le 1.$$

Thus, the inequality

$$\Vert {\displaystyle \frac{{\mathfrak{J}}^{\prime }(\top )}{{\top }^{-2}}}+1\Vert =\Vert \sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}{a}_{\mathfrak{n}}{\top }^{\mathfrak{n}+1}\Vert \le \sum _{\mathfrak{n}=1}^{\infty }\mathfrak{n}{a}_{\mathfrak{n}}{\Vert \top \Vert }^{\mathfrak{n}+1}<1-\wp$$

holds true if

$${\displaystyle \frac{{\mathfrak{n}\Vert \top \Vert }^{\mathfrak{n}+1}}{1-\wp }}\le {\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12(\varsigma -1)}}.$$

(38)

Then, inequality (38) holds true if

$${\Vert \top \Vert }^{\mathfrak{n}+1}\le {\displaystyle \frac{(1-\wp )\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12\mathfrak{n}(\varsigma -1)}}.$$

This leads to the close-to-convexity of the family and concludes the proof. □

Theorem 10.

Let $\mathfrak{J}\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. Then, $\mathfrak{J}$ is meromorphically starlike of order ℘ (where $0\le \wp <1$) in the disk $\left|z\right|<r_2$, where

$$r_2=\underset{\mathfrak{n}\ge 1}{inf}{\left({\displaystyle \frac{1-\wp }{\mathfrak{n}+2-\wp }}{\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12(\varsigma -1)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{n}+1}}}.$$

The result is exact for the extremal function $\mathfrak{J}$ defined by Equation (11).

Proof. 

Using the method applied in the proof of Theorem 2, it can be shown that

$$\Vert {\displaystyle \frac{\top {\mathfrak{J}}^{\prime }(\top )}{\mathfrak{J}(\top )}}+1\Vert <1-\wp ,\mathrm{for}\left|z\right|<r_2.$$

□

Theorem 11.

Let $\mathfrak{J}\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. Then, $\mathfrak{J}$ is meromorphically convex of order ℘ (where $0\le \wp <1$) in the disk $\left|z\right|<r_3$, where

$$r_3=\underset{\mathfrak{n}\ge 1}{inf}{\left({\displaystyle \frac{1-\wp }{\mathfrak{n}+2-\wp }}{\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12\mathfrak{n}(\varsigma -1)}}\right)}^{{\displaystyle \frac{1}{\mathfrak{n}+1}}}.$$

The outcome holds with the equality for the extremal function $\mathfrak{J}$ described in Equation (11).

Proof. 

By using the technique employed in the proof of Theorem 2, it can be demonstrated that

$$\Vert {\displaystyle \frac{\top {\mathfrak{J}}^{″}(\top )}{{\mathfrak{J}}^{\prime }(\top )}}+2\Vert <1-\wp ,\mathrm{for}\left|z\right|<r_2,$$

and establish that the theorem statement is accurate. □

8. Hadamard Product

Theorem 12.

For functions $\mathfrak{J},\mathfrak{K}\in {\mathsf{\Sigma }}_p$ defined by Equations (1) and (2), let $\mathfrak{J},\mathfrak{K}\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. Then, the Hadamard product $\mathfrak{J}\ast \mathfrak{K}\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$, where

$$\mathcal{E}\le {\displaystyle \frac{\mathfrak{n}{[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]}^2\psi (\mathfrak{n},\mu ,\vartheta )+{(1-\varsigma )}^2[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋]}{2⋋{(1-\varsigma )}^2(\mathfrak{n}-1)+\mathfrak{n}{[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]}^2\psi (\mathfrak{n},\mu ,\vartheta )}}$$

Proof. 

From relations (15) and (16), we need to find the largest $\mathcal{E}$ such that

$$\sum _{\mathfrak{n}=1}^{\infty }{\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\mathcal{E}-2\mathcal{E}+1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12(\mathcal{E}-1)}}{a}_{\mathfrak{n}}{b}_{\mathfrak{n}}\le 1.$$

(39)

From inequalities (10) and (16), it follows, through the application of the Cauchy–Schwarz inequality, that

$$\sum _{\mathfrak{n}=1}^{\infty }{\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12(\varsigma -1)}}\sqrt{a_{\mathfrak{n}}{b}_{\mathfrak{n}}}\le 1$$

(40)

Thus, it is enough to show that

$$\sqrt{a_{\mathfrak{n}}{b}_{\mathfrak{n}}}\le {\displaystyle \frac{(\mathcal{E}-1)[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]}{(\varsigma -1)[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\mathcal{E}-2\mathcal{E}+1)]}}.$$

(41)

On the other hand, from Equation (40), it is obtained that

$$\sqrt{a_{\mathfrak{n}}{b}_{\mathfrak{n}}}\le {\displaystyle \frac{12(\varsigma -1)}{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}}.$$

(42)

Therefore, in view of Equations (41) and (42), it is enough to show that

$${\displaystyle \frac{12(\varsigma -1)}{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}}\le {\displaystyle \frac{(\mathcal{E}-1)[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]}{(\varsigma -1)[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\mathcal{E}-2\mathcal{E}+1)]}},$$

which simplifies to

$$\mathcal{E}\le {\displaystyle \frac{\mathfrak{n}{[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]}^2\psi (\mathfrak{n},\mu ,\vartheta )+{(1-\varsigma )}^2[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋]}{2⋋{(1-\varsigma )}^2(\mathfrak{n}-1)+\mathfrak{n}{[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]}^2\psi (\mathfrak{n},\mu ,\vartheta )}}=\phi \left(\mathfrak{n}\right).$$

A straightforward calculation confirms that $\phi (\mathfrak{n}+1)-\phi \left(\mathfrak{n}\right)>0$ for all $\phi \left(\mathfrak{n}\right)$. This implies that $\phi \left(\mathfrak{n}\right)$ is an increasing function and satisfies $\phi \left(\mathfrak{n}\right)\ge \phi \left(1\right)$. Hence, the conclusion follows. □

Theorem 13.

For functions $\mathfrak{J},\mathfrak{K}\in {\mathsf{\Sigma }}_p$ defined by (1) and (2), let $\mathfrak{J},\mathfrak{K}\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. Then, the function $k\left(z\right)={\displaystyle \frac{1}{z}}+{\sum }_{n=1}^{\infty }(a_{\mathfrak{n}}^2+b_{\mathfrak{n}}^2)z^{\mathfrak{n}}$ is in the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$ and

$$\mathcal{E}\le {\displaystyle \frac{\mathfrak{n}{\left(2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)\right)}^2\psi (\mathfrak{n},\mu ,\vartheta )-24{(\varsigma -1)}^2[⋋-2({\mathfrak{n}}^2-4-3\mathfrak{n})]}{\mathfrak{n}{\left(2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)\right)}^2\psi (\mathfrak{n},\mu ,\vartheta )+48⋋(\mathfrak{n}-1){(1-\varsigma )}^2}}.$$

Proof. 

Since $\mathfrak{J},g\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$, it is obtained that

$$\sum _{\mathfrak{n}=1}^{\infty }{\left({\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12(\varsigma -1)}}{a}_{\mathfrak{n}}\right)}^2\le 1$$

(43)

and

$$\sum _{\mathfrak{n}=1}^{\infty }{\left({\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12(\varsigma -1)}}{b}_{\mathfrak{n}}\right)}^2\le 1.$$

(44)

Combining inequalities (43) and (44), the following is obtained:

$$\sum _{\mathfrak{n}=1}^{\infty }{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12(\varsigma -1)}}\right)}^2(a_{\mathfrak{n}}^2+b_{\mathfrak{n}}^2)\le 1$$

But, we need to find the largest $\mathcal{E}$ such that

$$\sum _{\mathfrak{n}=1}^{\infty }\left({\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\mathcal{E}-2\mathcal{E}+1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12(\mathcal{E}-1)}}\right)(a_{\mathfrak{n}}^2+b_{\mathfrak{n}}^2)\le 1.$$

(45)

The inequality (45) would hold if

$${\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\mathcal{E}-2\mathcal{E}+1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12(\mathcal{E}-1)}}\le {\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12(\varsigma -1)}}\right)}^2.$$

Then, it is obtained that

$$\mathcal{E}\le {\displaystyle \frac{\mathfrak{n}{\left(2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)\right)}^2\psi (\mathfrak{n},\mu ,\vartheta )-24{(\varsigma -1)}^2[⋋-2({\mathfrak{n}}^2-4-3\mathfrak{n})]}{\mathfrak{n}{\left(2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)\right)}^2\psi (\mathfrak{n},\mu ,\vartheta )+48⋋(\mathfrak{n}-1){(1-\varsigma )}^2}}.$$

A simple computation shows that $\psi (\mathfrak{n}+1)-\psi \left(\mathfrak{n}\right)>0$ for all $\mathfrak{n}$. This means that $\psi \left(\mathfrak{n}\right)$ is increasing and $\psi \left(\mathfrak{n}\right)\ge \psi \left(1\right)$. □

9. Integral Operators

In this section, we explore integral transforms of function in the family ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$, with a focus on the type of transforms discussed in [20].

Theorem 14.

Let the function $\mathfrak{J}\left(z\right)$ given by (1) be in the family ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. Then, the integral operator

$$F\left(z\right)=\mathfrak{C}{\int }_0^1{u}^{\mathfrak{C}}\mathfrak{J}\left(uz\right)du,0<u\le 1,0<\mathfrak{C}<\infty$$

(46)

is in the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$, where

$$\mathcal{E}={\displaystyle \frac{(2\mathfrak{n}\varsigma -2\varsigma +1)(\mathfrak{C}+\mathfrak{n}+1)+\mathfrak{C}(\varsigma -1)}{(2\mathfrak{n}\varsigma -2\varsigma +1)(\mathfrak{C}+\mathfrak{n}+1)-2\mathfrak{C}(\varsigma -1)(\mathfrak{n}-1)}}.$$

(47)

Proof. 

Let $\mathfrak{J}\in {\mathsf{\Sigma }}_p$ be given by (1) in the class ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$. Then,

$$F\left(z\right)=\mathfrak{C}{\int }_0^1{u}^{\mathfrak{C}}\mathfrak{J}\left(uz\right)du={\displaystyle \frac{1}{z}}+\sum _{\mathfrak{n}=1}^{\infty }{\displaystyle \frac{\mathfrak{C}}{\mathfrak{C}+\mathfrak{n}+1}}{a}_{\mathfrak{n}}{z}^{\mathfrak{n}}.$$

To show that

$$\sum _{\mathfrak{n}=1}^{\infty }{\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\mathcal{E}-2\mathcal{E}+1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12(\mathfrak{C}+\mathfrak{n}+1)(\mathcal{E}-1)}}{a}_{\mathfrak{n}}\le 1$$

(48)

since $\mathfrak{J}\in {\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top )$, the following inequality holds:

$$\sum _{\mathfrak{n}=1}^{\infty }{\displaystyle \frac{\mathfrak{n}[2({\mathfrak{n}}^2-4-3\mathfrak{n})-⋋(2\mathfrak{n}\varsigma -2\varsigma +1)]\psi (\mathfrak{n},\mu ,\vartheta )}{12(\varsigma -1)}}{a}_{\mathfrak{n}}\le 1.$$

(49)

The inequality is satisfied if

$${\displaystyle \frac{-\mathfrak{C}[2\mathfrak{n}\mathcal{E}-2\mathcal{E}+1)}{(1-\mathcal{E})(\mathfrak{C}+\mathfrak{n}+1)}}\le {\displaystyle \frac{-(2\mathfrak{n}\varsigma -2\varsigma +1)}{\varsigma -1}}$$

(50)

Thus, it follows that

$$\mathcal{E}\le {\displaystyle \frac{(2\mathfrak{n}\varsigma -2\varsigma +1)(\mathfrak{C}+\mathfrak{n}+1)+\mathfrak{C}(\varsigma -1)}{(2\mathfrak{n}\varsigma -2\varsigma +1)(\mathfrak{C}+\mathfrak{n}+1)-2\mathfrak{C}(\varsigma -1)(\mathfrak{n}-1)}}.$$

(51)

Let

$$\varphi \left(\mathfrak{n}\right)={\displaystyle \frac{(2\mathfrak{n}\varsigma -2\varsigma +1)(\mathfrak{C}+\mathfrak{n}+1)+\mathfrak{C}(\varsigma -1)}{(2\mathfrak{n}\varsigma -2\varsigma +1)(\mathfrak{C}+\mathfrak{n}+1)-2\mathfrak{C}(\varsigma -1)(\mathfrak{n}-1)}}.$$

It can be shown that $\psi \left(\mathfrak{n}\right)$ is an increasing function for $\mathfrak{n}\ge 1$, and that $\psi \left(\mathfrak{n}\right)\ge \psi \left(1\right)$. Using this result, the desired inequality is obtained. □

10. Conclusions

In this paper, we introduced and analyzed a new subclass of meromorphic functions, denoted by ${\mathfrak{A}}_{\vartheta }^{\mu }(\varsigma ,⋋,\top ),$ associated with the Mittag–Leffler function $E_{\mu ,\vartheta }\left(z\right)$, within the punctured symmetric domain ${⋓}^{\ast }$. We established several analytic and geometric properties of this class, such as coefficient estimates, growth and distortion bounds, and radii of close-to-convexity, starlikeness, and convexity. Additionally, we demonstrated the closure of the class under the Hadamard product and derived integral mean inequalities.

The findings of this work do not merely generalize existing results in meromorphic function theory but also offer new directions in geometric function theory. This research can be extended in various ways. One potential direction is the application of different forms of the Mittag–Leffler function that have been extensively studied and generalized in the literature. For example, in 1971, Prabhakar [21] introduced a more general function $E_{\alpha ,\beta }^{\gamma }\left(z\right)$, now commonly referred to as the Prabhakar function. Another notable generalization is the so-called k-Mittag–Leffler function $E_k\left(z\right)$, introduced in [22], which has also gained attention in recent studies. These generalized functions, when incorporated with suitable differential and integral operators, may lead to new subclasses of meromorphic functions and deeper insights into their structural and geometric behavior.

Therefore, this study not only lays a foundation for further theoretical development but also invites future investigations into broader generalizations involving fractional calculus and complex analysis.