Convolution Results with Subclasses of p-Valent Meromorphic Function Connected with $\mathfrak{q}$-Difference Operator

Abstract

Applying the operator of $\mathfrak{q}$-difference, we examine the convolution properties of the subclasses ${\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{A},\mathcal{B})$ and $\mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{A},\mathcal{B})$ of p-valent meromorphic functions defined in the punctured open-unit disc. We derived specific inclusion features and coefficient estimates for functions that fall into these subclasses. Additionally, connections between the results presented here and those discovered in earlier papers are emphasized.

1. Introduction

$\mathfrak{q}$-calculus is a branch of mathematics that extends the concepts of calculus to include $\mathfrak{q}$-derivatives and $\mathfrak{q}$-integrals. In traditional calculus, the derivative measures the rate of change in a function at a single point, while the integral measures the accumulation of a quantity over an interval. $\mathfrak{q}$-calculus introduces a parameter $\mathfrak{q}$ that generalizes these concepts and allows for more flexibility in calculations. $\mathfrak{q}$-derivatives and $\mathfrak{q}$-integrals can be defined in terms of $\mathfrak{q}$-differences and $\mathfrak{q}$-sums, which are analogous to the traditional notions of differences and sums. This extension of calculus has applications in various fields such as physics, engineering, and economics, where non-classical behavior or quantum phenomena are present. The capability of $\mathfrak{q}$-calculus to handle non-commutative variables is one of its primary features, meaning that the order in which variables are multiplied or operated on can affect the result. This property is crucial in quantum mechanics and other areas of physics where operators do not necessarily commute. $\mathfrak{q}$-calculus also provides a way to interpolate between classical calculus and other mathematical frameworks, such as discrete calculus and quantum calculus. By incorporating $\mathfrak{q}$-derivatives and $\mathfrak{q}$-integrals into their work, researchers are able to model more complex systems and phenomena accurately. The use of $\mathfrak{q}$-analysis in science and mathematics has recently piqued the interest of academics Jackson [1,2], Carmichael [3], Mason [4], and Trjitzinsky [5]. Ismail et al. [6] defined the $\mathfrak{q}$-starlike function. Several researchers have studied different $\mathfrak{q}$-calculus applications for subclasses of analytic functions; see [7,8,9,10,11,12,13,14,15,16]. Several mathemations focused on the classes of $\mathfrak{q}$-starlike functions associated with the Janowski functions [17] in a recent sequence, albeit from various angles.

The class of p-valently meromorphic functions of the following type is indicated by ${\Sigma }_p$:

$$\mathfrak{f}\left(\epsilon \right)={\epsilon }^{-p}+\sum _{\kappa =1-p}^{\infty }{a}_{\kappa }{\epsilon }^{\kappa }(p\in \mathbb{N}=\{1,2,\dots \}),$$

(1)

which are analytic in the punctured unit disc ${\mathfrak{U}}^{*}=\{\epsilon :\epsilon \in \mathbb{C}$ and $0<\left|\epsilon \right|<1\}=\mathfrak{U}\setminus \{0\}$. For simplicity, we write ${\Sigma }_1=\Sigma$.

From [18,19,20], if $\mathfrak{f}$ and $\mathfrak{l}$ are analytic in $\mathfrak{U}$, $\mathfrak{f}$ is subordinate to $\mathfrak{l}$, denoted as $\mathfrak{f}\left(\epsilon \right)\prec \mathfrak{l}\left(\epsilon \right)$. If there is an analytic function $\varpi$, with $\varpi \left(0\right)=0$ and $\left|\varpi \left(\epsilon \right)\right|<1$ for all $\epsilon \in \mathfrak{U}$, such that $\mathfrak{f}\left(\epsilon \right)=\mathfrak{l}\left(\varpi \right(\epsilon \left)\right)$, $\epsilon \in \mathfrak{U}$. If the function $\mathfrak{l}$ is univalent in $\mathfrak{U}$, $\mathfrak{f}\left(\epsilon \right)\prec \mathfrak{l}\left(\epsilon \right)$ is given as

$$\mathfrak{f}\left(0\right)=\mathfrak{l}\left(0\right)\mathrm{and}\mathfrak{f}\left(\mathfrak{U}\right)\subset \mathfrak{l}\left(\mathfrak{U}\right).$$

For $\mathfrak{f}\in {\Sigma }_p$, written as (1), and $\chi \in$${\Sigma }_p$ as specified by

$$\chi \left(\epsilon \right)={\epsilon }^{-p}+\sum _{\kappa =1-p}^{\infty }{b}_{\kappa }{\epsilon }^{\kappa }(p\in \mathbb{N}),$$

the popular convolution product that is

$$(\mathfrak{f}\ast \chi )\left(\epsilon \right)={\epsilon }^{-p}+\sum _{\kappa =1-p}^{\infty }{a}_{\kappa }{b}_{\kappa }{\epsilon }^{\kappa }=(\chi \ast \mathfrak{f})\left(\epsilon \right).$$

(2)

Now, we define two subclasses $\Sigma {\mathcal{S}}^p[\mathcal{D},E]$ and $\Sigma {\mathfrak{K}}^p[\mathcal{D},E]$ of the class ${\Sigma }_p$, for $-1\le E<\mathcal{D}\le 1$ and $p\in \mathbb{N}$ as follows

$$\Sigma {\mathcal{S}}^p[\mathcal{D},E]=\left\{\mathfrak{f}\in {\Sigma }_p:-{\displaystyle \frac{\epsilon f^{\prime }\left(\epsilon \right)}{p\mathfrak{f}\left(\epsilon \right)}}\prec {\displaystyle \frac{1+\mathcal{D}\epsilon }{1+E\epsilon }},\epsilon \in {\mathfrak{U}}^{*}\right\}$$

and

$$\Sigma {\mathfrak{K}}^p[\mathcal{D},E]=\left\{\mathfrak{f}\in {\Sigma }_p:-{\displaystyle \frac{1}{p}}\left(1+{\displaystyle \frac{\epsilon f^{″}\left(\epsilon \right)}{{\mathfrak{f}}^{\prime }\left(\epsilon \right)}}\right)\prec {\displaystyle \frac{1+\mathcal{D}\epsilon }{1+E\epsilon }},\epsilon \in {\mathfrak{U}}^{*}\right\}.$$

Clearly,

$$\mathfrak{f}\left(\epsilon \right)\in \Sigma {\mathfrak{K}}^p[\mathcal{D},E]\iff {\displaystyle \frac{-\epsilon f^{\prime }\left(\epsilon \right)}{p}}\in \Sigma {\mathcal{S}}^p[\mathcal{D},E].$$

We note that the class $\Sigma {\mathfrak{K}}^p[\mathcal{D},E]$ was studied by Mogra [21] and the class $\Sigma {\mathcal{S}}^p[\mathcal{D},E]$ was studied by Srivastava et al. [22].

Also, we note that

$$\Sigma {\mathcal{S}}^p[1-{\displaystyle \frac{2\varsigma }{p}},-1]=\Sigma {\mathcal{S}}^p\left(\varsigma \right),\Sigma {\mathfrak{K}}^p[1-{\displaystyle \frac{2\varsigma }{p}},-1]=\Sigma {\mathfrak{K}}^p\left(\varsigma \right),(0\le \varsigma <p),$$

where $\Sigma {\mathcal{S}}^p\left(\varsigma \right)$ and $\Sigma {\mathfrak{K}}^p\left(\varsigma \right)$ are, respectively, the classes of meromorphically p-valent starlike functions of order $\varsigma$ and meromorphically p-valent convex functions of order $\varsigma$$(0\le \varsigma <p)$ (see Aouf [23]).

The Jackson’s $\mathfrak{q}$-difference operator ${\mathfrak{D}}_{\mathfrak{q}}:{\Sigma }_p\to {\Sigma }_p$ is presented

$${\mathfrak{D}}_{\mathfrak{q}}\mathfrak{f}\left(\epsilon \right):=\left\{\begin{array}{cc}{\displaystyle \frac{\mathfrak{f}\left(\epsilon \right)-\mathfrak{f}\left(\mathfrak{q}\epsilon \right)}{(1-\mathfrak{q})\epsilon }}\hfill & \left(\epsilon \ne 0;0<\mathfrak{q}<1\right),\hfill \\ {\mathfrak{f}}^{\prime }\left(0\right)\hfill & \left(\epsilon =0\right),\hfill \\ {\mathfrak{f}}^{\prime }\left(\epsilon \right)\hfill & \left(\mathfrak{q}\to 1^{-},\epsilon \ne 0\right),\hfill \end{array}\right.$$

(3)

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

It comes to light that, for $\mathfrak{f}\in {\Sigma }_p,\kappa \in \mathbb{N}$ and $\epsilon \in {\mathfrak{U}}^{*},$

$$\underset{\mathfrak{q}\to 1^{-}}{lim}{\mathfrak{D}}_{\mathfrak{q}}\mathfrak{f}\left(\epsilon \right)=\underset{\mathfrak{q}\to 1^{-}}{lim}{\displaystyle \frac{\mathfrak{f}\left(\epsilon \right)-\mathfrak{f}\left(\mathfrak{q}\epsilon \right)}{(1-\mathfrak{q})\epsilon }}={\mathfrak{f}}^{\prime }\left(\epsilon \right).$$

By (1), we can see that

$${\mathfrak{D}}_{\mathfrak{q}}\mathfrak{f}\left(\epsilon \right)=-{\mathfrak{q}}^{-p}{\left[p\right]}_{\mathfrak{q}}{\epsilon }^{-1-p}+\sum _{\kappa =1-p}^{\infty }{\left[\kappa \right]}_{\mathfrak{q}}{a}_{\kappa }{\epsilon }^{\kappa -1}(\epsilon \ne 0),$$

where

$$\begin{array}{ccc}\hfill {\left[-p\right]}_{\mathfrak{q}}& =& {\displaystyle \frac{1-{\mathfrak{q}}^{-p}}{1-\mathfrak{q}}}=-{\mathfrak{q}}^{-p}{\left[p\right]}_{\mathfrak{q}}and\hfill \\ \hfill {\left[p\right]}_{\mathfrak{q}}& =& \left\{\begin{array}{cc}{\displaystyle \frac{1-{\mathfrak{q}}^p}{1-\mathfrak{q}}}& ifp\in \mathbb{C}\setminus \left\{0\right\}\\ {\mathfrak{q}}^{p-1}+{\mathfrak{q}}^{p-1}+{\mathfrak{q}}^{p-1}+\dots +\mathfrak{q}+1=\sum _{j=0}^{p-1}{\mathfrak{q}}^j& ifp\in \mathbb{N}\\ 1& if\mathfrak{q}\to 0^{+},p\in \mathbb{C}\setminus \left\{0\right\}\\ p& if\mathfrak{q}\to 1^{-},p\in \mathbb{C}\setminus \left\{0\right\}.\end{array}\right.\hfill \end{array}$$

Now, we defined the operator ${\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}:{\Sigma }_p\to {\Sigma }_p$ For $\mathfrak{f}\in {\Sigma }_p,\zeta \ge 0,p\in \mathbb{N},r\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ as follows:

$$\begin{array}{ccc}\hfill {\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{0,p}\mathfrak{f}\left(\epsilon \right)& =& \mathfrak{f}\left(\epsilon \right)\hfill \\ \hfill {\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{1,p}\mathfrak{f}\left(\epsilon \right)& =& \left(1-\zeta \right)\mathfrak{f}\left(\epsilon \right)+{\displaystyle \frac{\zeta }{{\epsilon }^p}}{\mathfrak{D}}_{\mathfrak{q}}\left({\epsilon }^{p+1}\mathfrak{f}\left(\epsilon \right)\right)\hfill \\ & :& \\ \hfill {\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)& =& \left(1-\zeta \right){\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r-1,p}\mathfrak{f}\left(\epsilon \right)+{\displaystyle \frac{\zeta }{{\epsilon }^p}}{\mathfrak{D}}_{\mathfrak{q}}\left({\epsilon }^{p+1}{\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r-1,p}\mathfrak{f}\left(\epsilon \right)\right)\hfill \end{array}$$

then

$${\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)={\epsilon }^{-p}+\sum _{\kappa =1-p}^{\infty }{\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa }.$$

(4)

From (4), we can obtain

$$\zeta {\mathfrak{q}}^{p+1}\epsilon {\partial }_{\mathfrak{q}}\left({\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)\right)={\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r+1,p}\mathfrak{f}\left(\epsilon \right)-\left(1+\zeta \mathfrak{q}{\left[p\right]}_{\mathfrak{q}}\right){\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)(\zeta >0).$$

(5)

The subordination principle between analytical functions is a fundamental concept in mathematics that helps to define relationships between different functions. Convergent power series are a representation of analytical functions, and the subordination principle states that if one function is analytically subordinate to another function in a certain domain, then the former is said to be subordinate to the latter in that domain. This principle is crucial in studying the behavior of complex function mappings and is often used in the field of complex analysis. Using the subordination principle, we introduce the subclasses ${\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$ and $\mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$:

$$\begin{array}{ccc}\hfill {\mathcal{MS}}_{\mathfrak{q}}^p(\mathcal{D},E)& =& \left\{\mathfrak{f}\in {\Sigma }_p:{\displaystyle \frac{-{\mathfrak{q}}^p\epsilon {\partial }_{\mathfrak{q}}\mathfrak{f}\left(\epsilon \right)}{{\left[p\right]}_{\mathfrak{q}}\mathfrak{f}\left(\epsilon \right)}}\prec {\displaystyle \frac{1+\mathcal{D}\epsilon }{1+E\epsilon }}\right\},\hfill \\ & & (0<\mathfrak{q}<1;-1\le E<\mathcal{D}\le 1;\epsilon \in \mathfrak{U}),\hfill \end{array}$$

(6)

and

$$\begin{array}{ccc}\hfill \mathcal{M}{\mathfrak{K}}_{\mathfrak{q}}^p(\mathcal{D},E)& =& \left\{\mathfrak{f}\in {\Sigma }_p:{\displaystyle \frac{-{\mathfrak{q}}^p{\partial }_{\mathfrak{q}}\left(\epsilon {\partial }_{\mathfrak{q}}\mathfrak{f}\left(\epsilon \right)\right)}{{\left[p\right]}_{\mathfrak{q}}{\partial }_{\mathfrak{q}}\mathfrak{f}\left(\epsilon \right)}}\prec {\displaystyle \frac{1+\mathcal{D}\epsilon }{1+E\epsilon }}\right\},\hfill \\ & & (0<\mathfrak{q}<1;-1\le E<\mathcal{D}\le 1;\epsilon \in \mathfrak{U}).\hfill \end{array}$$

(7)

Using (6) and (7), we have

$$\mathfrak{f}\left(\epsilon \right)\in \mathcal{M}{\mathfrak{K}}_{\mathfrak{q}}^p(\mathcal{D},E)\iff {\displaystyle \frac{-{\mathfrak{q}}^p\epsilon {\partial }_{\mathfrak{q}}\mathfrak{f}\left(\epsilon \right)}{{\left[p\right]}_{\mathfrak{q}}}}\in {\mathcal{MS}}_{\mathfrak{q}}^p(\mathcal{D},E).$$

(8)

Definition 1.

For $r\in {\mathbb{N}}_0$$0<\zeta <1,0<\mathfrak{q}<1;-1\le E<\mathcal{D}\le 1$ and $\epsilon \in \mathfrak{U}$, we introduce the following subclasses:

$${\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)=\left\{\mathfrak{f}\in {\Sigma }_p:{\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)\in {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)\right\},$$

(9)

and

$$\mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)=\left\{\mathfrak{f}\in {\Sigma }_p:{\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)\in \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)\right\}.$$

(10)

Using (9) and (10), we have

$${\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}f\left(\epsilon \right)\in \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)\iff {\displaystyle \frac{-{\mathfrak{q}}^p\epsilon {\partial }_{\mathfrak{q}}{\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)}{{\left[p\right]}_{\mathfrak{q}}}}\in {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E).$$

Remark 1.

(i) ${\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(1-{\displaystyle \frac{2\varsigma }{p}},-1)={\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}\left(\varsigma \right)$

$$\begin{array}{c}\hfill =\left\{\mathfrak{f}\in {\Sigma }_p:Re\left({\displaystyle \frac{-{\mathfrak{q}}^p\epsilon {\partial }_{\mathfrak{q}}{\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)}{{\left[p\right]}_{\mathfrak{q}}{\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)}}\right)>\varsigma ;0\le \varsigma <p,\epsilon \in \mathfrak{U}\right\}\end{array}$$

and

$$\begin{array}{ccc}\hfill \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(1-{\displaystyle \frac{2\varsigma }{p}},-1)& =& \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}\left(\varsigma \right)\hfill \\ & =& \left\{\mathfrak{f}\in {\Sigma }_p:Re\left({\displaystyle \frac{-{\mathfrak{q}}^p{\partial }_{\mathfrak{q}}\left(\epsilon {\partial }_{\mathfrak{q}}{\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)\right)}{{\left[p\right]}_{\mathfrak{q}}{\partial }_{\mathfrak{q}}{\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)}}\right)>\varsigma ,0\le \varsigma <1,\epsilon \in \mathfrak{U}\right\}\hfill \end{array}$$

are the subclasses of p-valent meromorphic $\mathfrak{q}$-starlike and $\mathfrak{q}$-convex functions of order ς, respectively;

(ii) ${\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,1}(1-2\varsigma ,-1)={\mathcal{MS}}_{\zeta ,\mathfrak{q}}^r\left(\varsigma \right)$

$$\begin{array}{c}\hfill =\left\{\mathfrak{f}\in \Sigma :Re\left({\displaystyle \frac{-\mathfrak{q}\epsilon {\partial }_{\mathfrak{q}}{\mathfrak{D}}_{\zeta ,\mathfrak{q}}^r\mathfrak{f}\left(\epsilon \right)}{{\mathfrak{D}}_{\zeta ,\mathfrak{q}}^r\mathfrak{f}\left(\epsilon \right)}}\right)>\varsigma ;0\le \varsigma <1,\epsilon \in \mathfrak{U}\right\}\end{array}$$

and

$$\begin{array}{ccc}\hfill \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,1}(1-2\varsigma ,-1)& =& \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^r\left(\varsigma \right)\hfill \\ & =& \left\{\mathfrak{f}\in \Sigma :Re\left({\displaystyle \frac{-\mathfrak{q}{\partial }_{\mathfrak{q}}\left(\epsilon {\partial }_{\mathfrak{q}}{\mathfrak{D}}_{\zeta ,\mathfrak{q}}^r\mathfrak{f}\left(\epsilon \right)\right)}{{\partial }_{\mathfrak{q}}{\mathfrak{D}}_{\zeta ,\mathfrak{q}}^r\mathfrak{f}\left(\epsilon \right)}}\right)>\varsigma ,0\le \varsigma <1,\epsilon \in \mathfrak{U}\right\}\hfill \end{array}$$

are the subclasses of univalent meromorphic $\mathfrak{q}$-starlike and $\mathfrak{q}$-convex functions of order ς, respectively;

(iii) ${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{0,p}(\mathcal{D},E)={\mathcal{MS}}^p(\mathcal{D},E)$

$$\begin{array}{c}\hfill =\left\{\mathfrak{f}\in {\Sigma }_p:-{\displaystyle \frac{\epsilon f^{\prime }\left(\epsilon \right)}{p\mathfrak{f}\left(\epsilon \right)}}\prec {\displaystyle \frac{1+\mathcal{D}\epsilon }{1+E\epsilon }};-1\le E<\mathcal{D}\le 1,\epsilon \in \mathfrak{U}\right\}\end{array}$$

and

$$\begin{array}{ccc}\hfill \underset{\mathfrak{q}\to 1^{-}}{lim}\mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{0,p}(\mathcal{D},E)& =& \mathcal{M}{\mathfrak{K}}^p(\mathcal{D},E)\hfill \\ & =& \left\{\mathfrak{f}\in {\Sigma }_p:-{\displaystyle \frac{1}{p}}\left(1+{\displaystyle \frac{\epsilon f^{″}\left(\epsilon \right)}{{\mathfrak{f}}^{\prime }\left(\epsilon \right)}}\right)\prec {\displaystyle \frac{1+\mathcal{D}\epsilon }{1+E\epsilon }};-1\le E<\mathcal{D}\le 1,\epsilon \in \mathfrak{U}\right\}\hfill \\ \end{array}$$

see [24].

(iv) ${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{MS}}_{1,\mathfrak{q}}^{0,1}(1-2\varsigma ,-1)=\mathcal{MS}\left(\varsigma \right)$

$$\begin{array}{c}\hfill =\left\{\mathfrak{f}\in \Sigma :Re\left(-{\displaystyle \frac{\epsilon f^{\prime }\left(\epsilon \right)}{\mathfrak{f}\left(\epsilon \right)}}\right)>\varsigma ;0\le \varsigma <1,\epsilon \in \mathfrak{U}\right\}\end{array}$$

and

$$\begin{array}{ccc}\hfill \underset{\mathfrak{q}\to 1^{-}}{lim}\mathcal{M}{\mathfrak{K}}_{1,\mathfrak{q}}^{0,1}(1-2\varsigma ,-1)& =& \mathcal{M}\mathfrak{K}\left(\varsigma \right)\hfill \\ & =& \left\{\mathfrak{f}\in \Sigma :Re\left(-1-{\displaystyle \frac{\epsilon f^{″}\left(\epsilon \right)}{{\mathfrak{f}}^{\prime }\left(\epsilon \right)}}\right)>\varsigma ,0\le \varsigma <1,\epsilon \in \mathfrak{U}\right\}\hfill \end{array}$$

were investigated by Kaczmarski [25];

(v) ${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{MS}}_{1,\mathfrak{q}}^{0,1}(1,-1)=\mathcal{MS}$ and ${lim}_{\mathfrak{q}\to 1^{-}}\mathcal{M}{\mathfrak{K}}_{1,\mathfrak{q}}^{0,1}(1,-1)=\mathcal{M}\mathfrak{K}$ which are well-known classes of starlike and convex univalent meromorphic functions, respectively, see [26,27,28].

Convolution features are an important aspect of several families of analytic functions. Convolution is a mathematical operation in which two functions are taken and a third function is produced. It is a fundamental operation in several areas of mathematics, including signal processing, image processing, and probability theory. In the context of analytic functions, convolution can be used to combine two functions to produce a new function that retains the properties of the original functions. Convolution features of several families of analytic and p-valent meromorphic functions have been the subject of a large body of scholarship; for specifics, see [29,30,31,32,33]. Now, we derive several convolution features of the meromorphic functions using the quantum derivative. We created the new classes ${\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$ and $\mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$ for this reason. Some implications, including necessary and sufficient criteria, coefficient estimates, and inclusion features of the subclasses ${\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$ and $\mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$, follow the convolution results.

2. Convolution Properties

This section will assume, unless otherwise stated, that $0<\mathfrak{q}<1;-1\le E<\mathcal{D}\le 1,\varphi \in \left[0,2\pi \right).$

Lemma 1

([34]). $\mathfrak{f}\in {\mathcal{MS}}_{\mathfrak{q}}^p(\mathcal{D},E)$ iff

$${\epsilon }^p\left[\mathfrak{f}\left(\epsilon \right)\ast {\displaystyle \frac{1+(\mathfrak{C}-\mathfrak{q})\epsilon }{{\epsilon }^p(1-\epsilon )(1-\mathfrak{q}\epsilon )}}\right]\ne 0(\epsilon \in \mathfrak{U}),$$

(11)

for all

$$\mathfrak{C}={\displaystyle \frac{E+e^{-i\varphi }}{{\left[p\right]}_{\mathfrak{q}}(\mathcal{D}-E)}}$$

(12)

and also $\mathfrak{C}=0$.

Letting $\mathfrak{q}\to 1^{-}$ in Lemma 1, we have

Corollary 1.

$\mathfrak{f}\in {\mathcal{MS}}^p(\mathcal{D},E)$ iff

$${\epsilon }^p\left[\mathfrak{f}\left(\epsilon \right)\ast {\displaystyle \frac{1+(\mathfrak{C}-1)\epsilon }{{\epsilon }^p{(1-\epsilon )}^2}}\right]\ne 0(\epsilon \in \mathfrak{U}),$$

for all

$$\mathfrak{C}={\displaystyle \frac{E+e^{-i\varphi }}{p(\mathcal{D}-E)}}$$

and also $\mathfrak{C}=0$.

Remark 2.

$\left(i\right)$ Putting $p=1$ in Corollary 1, we obtain the results obtained by Bulboaca et al. ([30], Theorem 1, with $b=1)$.

$\left(ii\right)$ Putting $E=\left(1-{\displaystyle \frac{2\varsigma }{p}}\right),$ and $\mathcal{D}=-1$$(0\le \varsigma <p),$ in Corollary 1, we give the necessary and sufficient condition for the function $\mathfrak{f}\in {\Sigma }_p$ to be in the class ${\mathcal{MS}}^p\left(\varsigma \right).$

Lemma 2

([34]). $\mathfrak{f}\in \mathcal{M}{\mathfrak{K}}_{\mathfrak{q}}^p(\mathcal{D},E)$ iff

$${\epsilon }^p\left[\mathfrak{f}\left(\epsilon \right)\ast {\displaystyle \frac{1-\frac{(1-{\mathfrak{q}}^{p+2})-\mathfrak{q}(1-{\mathfrak{q}}^{p-1})(\mathfrak{C}-\mathfrak{q})}{1-{\mathfrak{q}}^p}\epsilon -\frac{\mathfrak{q}(1-{\mathfrak{q}}^{p+1})(\mathfrak{C}-\mathfrak{q})}{1-{\mathfrak{q}}^p}{\epsilon }^2}{{\epsilon }^p(1-\epsilon )(1-\mathfrak{q}\epsilon )(1-{\mathfrak{q}}^2\epsilon )}}\right]\ne 0(\epsilon \in \mathfrak{U}),$$

(13)

for all $\mathfrak{C}$ defined by (12), and also $\mathfrak{C}=0$.

Letting $\mathfrak{q}\to 1^{-}$ in Lemma 2, we have

Corollary 2.

$\mathfrak{f}\in \mathcal{M}{\mathfrak{K}}^p(\mathcal{D},E)$ iff

$${\epsilon }^p\left[\mathfrak{f}\left(\epsilon \right)\ast \left\{{\displaystyle \frac{p-\{2+p-(p-1)(\mathfrak{C}-1)\}\epsilon -(p+1)(\mathfrak{C}-1){\epsilon }^2}{{\epsilon }^p{(1-\epsilon )}^3}}\right\}\right]\ne 0(\epsilon \in \mathfrak{U}),$$

for all

$$\mathfrak{C}={\displaystyle \frac{E+e^{-i\varphi }}{p(\mathcal{D}-E)}}$$

and also $\mathfrak{C}=0$.

Remark 3.

$\left(i\right)$ Putting $p=1$ in Corollary 2, we obtain the results obtained by Bulboaca et al. ([30], Theorem 2, with $b=1$).

$\left(ii\right)$ Putting $E=\left(1-{\displaystyle \frac{2\varsigma }{p}}\right),$ and $\mathcal{D}=-1$$(0\le \varsigma <p)$, in Corollary 2, we give the necessary and sufficient condition for the function $\mathfrak{f}\in {\Sigma }_p$ to be in the class $\mathcal{M}{\mathfrak{K}}^p\left(\varsigma \right).$

Theorem 1.

A neccessary and sufficient condition for $\mathfrak{f}\in {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$ is

$$1+\sum _{\kappa =1-p}^{\infty }{\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}\ne 0,$$

or

$$1+\sum _{\kappa =1-p}^{\infty }({[\kappa +p]}_{\mathfrak{q}}\mathfrak{C}+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}\ne 0(\epsilon \in \mathfrak{U}),$$

(14)

where $\mathfrak{C}$ is defined by (12).

Proof. 

From Lemma 1, we find that $\mathfrak{f}\in {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$ iff

$${\epsilon }^p\left[({\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f})\left(\epsilon \right)\ast {\displaystyle \frac{1+(\mathfrak{C}-\mathfrak{q})\epsilon }{{\epsilon }^p(1-\epsilon )(1-\mathfrak{q}\epsilon )}}\right]\ne 0(\epsilon \in \mathfrak{U}),$$

(15)

for all $\mathfrak{C}={\displaystyle \frac{E+e^{-i\varphi }}{{\left[p\right]}_{\mathfrak{q}}(\mathcal{D}-E)}}$ and also $\mathfrak{C}=0.$ We can write the left-hand side of (14) as follows:

$$\begin{array}{ccc}& & {\epsilon }^p\left[({\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f})\left(\epsilon \right)\ast {\displaystyle \frac{1+(\mathfrak{C}-\mathfrak{q})\epsilon }{{\epsilon }^p(1-\epsilon )(1-\mathfrak{q}\epsilon )}}\right]\hfill \\ & =& {\epsilon }^p\left[({\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f})\left(\epsilon \right)\ast \left({\displaystyle \frac{1-\mathfrak{q}\epsilon }{{\epsilon }^p(1-\epsilon )(1-\mathfrak{q}\epsilon )}}+{\displaystyle \frac{\mathfrak{C}\epsilon }{{\epsilon }^p(1-\epsilon )(1-\mathfrak{q}\epsilon )}}\right)\right]\hfill \\ & =& {\epsilon }^p\left\{({\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f})\left(\epsilon \right)\ast \left({\epsilon }^{-p}+\sum _{\kappa =1-p}^{\infty }({[\kappa +p]}_{\mathfrak{q}}\mathfrak{C}+1){\epsilon }^{\kappa }\right)\right\}\hfill \\ & =& 1+\sum _{\kappa =1-p}^{\infty }({[\kappa +p]}_{\mathfrak{q}}\mathfrak{C}+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}.\hfill \end{array}$$

Thus, we completed the proof. □

Theorem 2.

A neccessary and sufficient condition for $\mathfrak{f}\in \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$ is

$$1-\sum _{\kappa =1-p}^{\infty }{\displaystyle \frac{{\mathfrak{q}}^p}{{\left[p\right]}_{\mathfrak{q}}}}{\left[\kappa \right]}_{\mathfrak{q}}{\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}\ne 0(\epsilon \in \mathfrak{U}),$$

(16)

or

$$1-\sum _{\kappa =1-p}^{\infty }{\displaystyle \frac{{\mathfrak{q}}^p}{{\left[p\right]}_{\mathfrak{q}}}}{\left[\kappa \right]}_{\mathfrak{q}}({[\kappa +p]}_{\mathfrak{q}}\mathfrak{C}+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}\ne 0(\epsilon \in \mathfrak{U}).$$

(17)

Proof. 

From Lemma 2, we find that $\mathfrak{f}\in \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$ iff

$${\epsilon }^p\left[({\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f})\left(\epsilon \right)\ast {\displaystyle \frac{1-\frac{(1-{\mathfrak{q}}^{p+2})-\mathfrak{q}(1-{\mathfrak{q}}^{p-1})(\mathfrak{C}-\mathfrak{q})}{1-{\mathfrak{q}}^p}\epsilon -\frac{\mathfrak{q}(1-{\mathfrak{q}}^{p+1})(\mathfrak{C}-\mathfrak{q})}{1-{\mathfrak{q}}^p}{\epsilon }^2}{{\epsilon }^p(1-\epsilon )(1-\mathfrak{q}\epsilon )(1-{\mathfrak{q}}^2\epsilon )}}\right]\ne 0(\epsilon \in \mathfrak{U}),$$

(18)

for all $\mathfrak{C}={\displaystyle \frac{E+e^{-i\varphi }}{{\left[p\right]}_{\mathfrak{q}}(\mathcal{D}-E)}}$ and also $\mathfrak{C}=0.$ We can write the left-hand side of (17) as follows:

$$\begin{array}{ccc}& & {\epsilon }^p\left[({\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f})\left(\epsilon \right)\ast {\displaystyle \frac{1-\frac{(1-{\mathfrak{q}}^{p+2})-\mathfrak{q}(1-{\mathfrak{q}}^{p-1})(\mathfrak{C}-\mathfrak{q})}{1-{\mathfrak{q}}^p}\epsilon -\frac{\mathfrak{q}(1-{\mathfrak{q}}^{p+1})(\mathfrak{C}-\mathfrak{q})}{1-{\mathfrak{q}}^p}{\epsilon }^2}{{\epsilon }^p(1-\epsilon )(1-\mathfrak{q}\epsilon )(1-{\mathfrak{q}}^2\epsilon )}}\right]\hfill \\ & =& {\epsilon }^p\left[({\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f})\left(\epsilon \right)\ast \left({\displaystyle \frac{1}{{\epsilon }^p(1-\epsilon )(1-\mathfrak{q}\epsilon )(1-{\mathfrak{q}}^2\epsilon )}}-{\displaystyle \frac{\frac{(1-{\mathfrak{q}}^{p+2})-\mathfrak{q}(1-{\mathfrak{q}}^{p-1})(\mathfrak{C}-\mathfrak{q})}{1-{\mathfrak{q}}^p}\epsilon }{{\epsilon }^p(1-\epsilon )(1-\mathfrak{q}\epsilon )(1-{\mathfrak{q}}^2\epsilon )}}\right.\right.\hfill \\ & & \left.+\left.{\displaystyle \frac{\frac{\mathfrak{q}(1-{\mathfrak{q}}^{p+1})(\mathfrak{C}-\mathfrak{q})}{1-{\mathfrak{q}}^p}{\epsilon }^2}{{\epsilon }^p(1-\epsilon )(1-\mathfrak{q}\epsilon )(1-{\mathfrak{q}}^2\epsilon )}}\right)\right]\hfill \\ & =& 1-\sum _{\kappa =1-p}^{\infty }{\displaystyle \frac{{\mathfrak{q}}^p}{{\left[p\right]}_{\mathfrak{q}}}}{\left[\kappa \right]}_{\mathfrak{q}}({[\kappa +p]}_{\mathfrak{q}}\mathfrak{C}+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}.\hfill \end{array}$$

Thus, we completed the proof. □

Remark 4.

Putting $\zeta =1$ in the above results, we improve all results obtained by El-Qadeem and Elshazly (Theorem 4 [34]).

3. Coefficient Estimates and Inclusion Properties

As an application of Theorems 1 and 2, we will decide the coefficient estimate and inclusion results for a function given by (1), which belongs to ${\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$ and $\mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E).$

Theorem 3.

$\mathfrak{f}$ given by (1) satisfies

$$\sum _{\kappa =1-p}^{\infty }{\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r\left|a_{\kappa }\right|<1,$$

(19)

and

$$\sum _{\kappa =1-p}^{\infty }({[\kappa +p]}_{\mathfrak{q}}\left|\mathfrak{C}\right|+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r\left|a_{\kappa }\right|<1,$$

(20)

then $\mathfrak{f}\in {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E).$

Proof. 

Since

$$\begin{array}{ccc}& & \left|1+\sum _{\kappa =1-p}^{\infty }{\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}\right|\hfill \\ & \ge & 1-\left|\sum _{\kappa =1-p}^{\infty }{\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}\right|\hfill \\ & \ge & 1-\sum _{\kappa =1-p}^{\infty }{\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r\left|a_{\kappa }\right|\left|{\epsilon }^{\kappa +p}\right|\hfill \\ & >& 1-\sum _{\kappa =1-p}^{\infty }{\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r\left|a_{\kappa }\right|>0,\hfill \end{array}$$

which leads to (16), then $\mathfrak{f}\in {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$. Also, using (20), we obtain

$$\begin{array}{ccc}& & \left|1+\sum _{\kappa =1-p}^{\infty }({[\kappa +p]}_{\mathfrak{q}}\mathfrak{C}+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}\right|\hfill \\ & \ge & 1-\left|\sum _{\kappa =1-p}^{\infty }({[\kappa +p]}_{\mathfrak{q}}\mathfrak{C}+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}\right|\hfill \\ & \ge & 1-\sum _{\kappa =1-p}^{\infty }({[\kappa +p]}_{\mathfrak{q}}\mathfrak{C}+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r\left|a_{\kappa }\right|\left|{\epsilon }^{\kappa +p}\right|\hfill \\ & >& 1-\sum _{\kappa =1-p}^{\infty }({[\kappa +p]}_{\mathfrak{q}}\mathfrak{C}+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r\left|a_{\kappa }\right|>0,\hfill \end{array}$$

which shows that (17) holds true and $\mathfrak{f}\in {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$; the proof is completed. □

Now, we can prove the theorem below:

Theorem 4.

$\mathfrak{f}$ given by (1) satisfies

$$\sum _{\kappa =1-p}^{\infty }{\displaystyle \frac{{\mathfrak{q}}^p}{{\left[p\right]}_{\mathfrak{q}}}}{\left[\kappa \right]}_{\mathfrak{q}}{\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r\left|a_{\kappa }\right|<1,$$

and

$$\sum _{\kappa =1-p}^{\infty }{\displaystyle \frac{{\mathfrak{q}}^p}{{\left[p\right]}_{\mathfrak{q}}}}{\left[\kappa \right]}_{\mathfrak{q}}({[\kappa +p]}_{\mathfrak{q}}\left|\mathfrak{C}\right|+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r\left|a_{\kappa }\right|<1,$$

then $\mathfrak{f}\in \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E).$

Remark 5.

Putting $\zeta =1$ in the above results, we improve all results obtained by El-Qadeem and Elshazly (Theorem 6 [34]).

Theorem 5.

If $r\in {\mathbb{N}}_0,$ then ${\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r+1,p}(\mathcal{D},E)\subset {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E).$

Proof. 

If $\mathfrak{f}\in {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r+1,p}(\mathcal{D},E),$ then Theorem 3 gives

$$1+\sum _{\kappa =1-p}^{\infty }{\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^{r+1}{a}_{\kappa }{\epsilon }^{\kappa +p}\ne 0,$$

(21)

or

$$1+\sum _{\kappa =1-p}^{\infty }({[\kappa +p]}_{\mathfrak{q}}\mathfrak{C}+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^{r+1}{a}_{\kappa }{\epsilon }^{\kappa +p}\ne 0(\epsilon \in \mathfrak{U}),$$

(22)

but (21) and (22) can be written as follows:

$$\left(1+\sum _{\kappa =1-p}^{\infty }\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]{\epsilon }^{\kappa +p}\right)\ast \left(1+\sum _{\kappa =1-p}^{\infty }{\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}\right)\ne 0,$$

(23)

and

$$\begin{array}{c}\left(1+\sum _{\kappa =1-p}^{\infty }\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]{\epsilon }^{\kappa +p}\right)\hfill \\ \ast \left(1+\sum _{\kappa =1-p}^{\infty }({[\kappa +p]}_{\mathfrak{q}}\left|\mathfrak{C}\right|+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}\right)\ne 0.\hfill \end{array}$$

(24)

Let us really define the function

$$\omega \left(\epsilon \right)=1+\sum _{\kappa =1-p}^{\infty }\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]{\epsilon }^{\kappa +p}.$$

(25)

We note that the assumption $\omega \left(\epsilon \right)=0$ leads to $\left|\epsilon \right|>1$. Thus, we deduce that $\omega \left(\epsilon \right)\ne 0$. Using the property that if $\omega \ast \chi \ne 0$ and $\omega \ne 0$, then $\chi \ne 0$. Thus, from (23) and (24) and using the function $\omega \left(\epsilon \right)\ne 0$, we obtain

$$1+\sum _{\kappa =1-p}^{\infty }{\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}\ne 0,$$

and

$$1+\sum _{\kappa =1-p}^{\infty }({[\kappa +p]}_{\mathfrak{q}}\mathfrak{C}+1){\left[1+\zeta ({[\kappa +p+1]}_{\mathfrak{q}}-1)\right]}^r{a}_{\kappa }{\epsilon }^{\kappa +p}\ne 0,$$

then Theorem 1 tells us that $\mathfrak{f}\in {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E).$ □

Finally, we can prove the following theorem.

Theorem 6.

If $r\in {\mathbb{N}}_0,$ then $\mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r+1,p}(\mathcal{D},E)\subset \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E).$

Corollary 3.

Let $\nu =r+1,r+2,\dots (r\in {\mathbb{N}}_0).$ Then,

$$\mathfrak{f}\in {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{\nu ,p}(\mathcal{D},E)\Rightarrow \mathfrak{f}\in {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E).$$

Equivalently, if

$${\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)\in {\mathcal{MS}}_{\mathfrak{q}}^p(\mathcal{D},E),$$

then

$$\mathfrak{f}\in {\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E).$$

Corollary 4.

Let $\nu =r+1,r+2,\dots (r\in {\mathbb{N}}_0).$ Then,

$$\mathfrak{f}\in \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{\nu ,p}(\mathcal{D},E)\Rightarrow \mathfrak{f}\in \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E).$$

Equivalently, if

$${\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}\mathfrak{f}\left(\epsilon \right)\in \mathcal{M}{\mathfrak{K}}_{\mathfrak{q}}^p(\mathcal{D},E),$$

then

$$\mathfrak{f}\in \mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E).$$

Remark 6.

Putting $\zeta =1$ in Theorem 5 and Theorem 6, we retrieve the results obtained by El-Qadeem and Elshazly (Theorem 7, Theorem 8 [34]).

4. Conclusions

We introduced the new subclasses ${\mathcal{MS}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$ and $\mathcal{M}{\mathfrak{K}}_{\zeta ,\mathfrak{q}}^{r,p}(\mathcal{D},E)$ with the aid of the operator ${\mathfrak{D}}_{\zeta ,\mathfrak{q}}^{r,p}$. The convolution features of several families of analytic multivalued functions provide valuable insights into the behavior and properties of these functions in the complex plane. By exploring the convolution features of these families of functions, mathematicians can deepen their understanding of complex analysis and uncover new avenues for research and exploration in the realm of multivalued functions. Convolution circumstances were the main focus of the investigation. For future investigations, we recommend applying these subclasses in research on differential subordination and superordination theories in the future. Furthermore, the Fekete–Szego functionals can be obtained by defining the outcomes of the computation of the bi-univalent functions’ coefficient bounds $\mathfrak{f}\left(\zeta \right)$.