Sharp Results and Fluid Flow Applications for a Specific Class of Meromorphic Functions Introduced by a New Operator

Abstract

In this investigation, we introduce a new meromorphic operator defined by meromorphic univalent functions. A new class of meromorphic functions is introduced by this operator, which can generate several distinct subclasses depending on the values of its parameters. Within the framework of this class of functions, we obtain several significant algebraic and geometric properties, including coefficient estimates, distortion theorems, the radius of starlikeness, convex combination closure, extreme point characterization, and neighborhood structure. Our findings are sharp, offering accurate and significant insights into the mathematical structure and behavior of these functions. In addition, we present several applications of these results in fluid mechanics, like identifying stagnation points in vortex flows, predicting velocity decline in source/sink systems, and determining stability thresholds that protect crucial streamlines from perturbations, which demonstrates that the introduced operator and class characterize critical properties of 2D potential flows.

1. Introduction

The study of meromorphic functions began in the 19th century as an essential area of complex analysis, pioneered by Karl Weierstrass [1], who presented the canonical product representation and formally defined functions with isolated poles. Simultaneously, Bernhard Riemann [2] worked on Riemann surfaces and conformal mappings and established a geometric foundation for understanding singularities and multivalued behavior in complex functions, leading to an accurate definition of meromorphic functions as those that are analytic except at isolated poles. These functions generalized rational and entire functions, becoming pivotal in the theory of analytic continuation, residue calculus, and conformal mapping. Subsequently, geometric subclasses including meromorphic starlike and meromorphic convex functions were introduced to explore geometric function theory within punctured domains. These subclasses were introduced and extensively examined by Ch. Pommerenke [3] and further expanded by Miller in [4]. These classes’ geometric structure makes them more essential since it enables applications in univalence conformal invariants and geometric distortion. More recently, operators have been established to define and explore classes of meromorphic functions, allowing geometric bounds and coefficient estimates to be determined (see [5,6,7,8]). In geometric function theory, operators like the q-difference operator, which Jackson developed [9], have become indispensable tools that make it easier to create new subclasses and generalizations. Leading researchers in this field, such as Altintas and Owa [10], Srivastava [11], and Goodman [12], have made significant contributions by fusing operator theory with analytic techniques to derive sharp results on meromorphic univalent, starlike, and convex functions. These results remain crucial in both theoretical and applied complex analysis.

In addition to the theoretical development, we explore several practical applications of the introduced class and operator in fluid mechanics. Specifically, we demonstrate how important problems in modeling 2D potential flows with singularities are addressed by the operator and its related class. We prove the absence of stagnation spots under controlled perturbations for vortex interactions, provide rigorous velocity decay laws for source/sink systems, and construct stability thresholds that maintain streamlines for universal singularity-driven flows. These developments provide new mathematical foundations for forecasting and managing hydrodynamic behavior in punctured areas.

The paper is arranged as follows. In Section 2, we provide the essential definitions, notations, and mathematical preliminaries related to the introduced operator and class. Section 3 is devoted to developing the main theoretical results, including coefficient estimates, distortion theorems, starlikeness, convex combination closure, and extreme point characterizations, as well as studying the simple notion of the neighborhood. In Section 4, we present several applications of the proposed operator and function class in fluid mechanics. The paper is finally concluded in Section 5 with a summary and potential future research areas.

2. Mathematical Preliminaries

Denote by ∑ the class of univalent meromorphic functions:

$$F\left(z\right)={\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{a}_k{z}^k,$$

(1)

where $z\in U^{*}=\{z:z\in \mathbb{C}\mathrm{and}0<|z|<1\}$ and $a_k\ge 0$.

Definition 1.

For $F_j\left(z\right)\in \sum$$(j=1,2)$, given by

$$F_j\left(z\right)={\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{a}_{k,j}{z}^k(j=1,2),$$

(2)

$F_1\ast F_2$ is given by

$$(F_1\ast F_2)\left(z\right)={\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{a}_{k,1}{a}_{k,2}{z}^k=(F_2\ast F_1)\left(z\right).$$

(3)

Definition 2.

Pommerenke [3] introduced the classes of meromorphic starlike (${\sum }^{*}$) and meromorphic convex (${\sum }^c$) that satisfy

$$-\Re \left\{{\displaystyle \frac{z{F}^{\prime }\left(z\right)}{F\left(z\right)}}\right\}>0,$$

and

$$-\Re \left\{1+{\displaystyle \frac{z{F}^{\prime \prime }\left(z\right)}{F^{\prime }\left(z\right)}}\right\}>0.$$

Following that, Miller [4] introduced the classes of meromorphic starlike of order α (${\sum }^{*}\left(\alpha \right)$), and convex of order α (${\sum }^c\left(\alpha \right)$) that satisfy

$$-\Re \left\{{\displaystyle \frac{z{F}^{\prime }\left(z\right)}{F\left(z\right)}}\right\}>\alpha ,$$

(4)

and

$$-\Re \left\{1+{\displaystyle \frac{z{F}^{\prime \prime }\left(z\right)}{F^{\prime }\left(z\right)}}\right\}>\alpha ,(0\le \alpha <1).$$

(5)

These classes have been investigated by Kumar and Shukla [13], Uralegaddi, Somanatha [14], and many other authors (see [15,16,17]).

The q-derivative operator was introduced by Jackson [9]. Gasper and Rahman [18] extended this operator to meromorphic functions in class ∑, establishing its convergence and pole handling properties.

Definition 3.

Gasper and Rahman [18] defined the operator ${\nabla }_q$ for $F\left(z\right)\in \sum$, and $0<q<1$ as

$${\nabla }_{q}F\left(z\right)={\displaystyle \frac{F\left(z\right)-F\left(qz\right)}{(1-q)z}}\mathit{for}z\ne 0,$$

that is

$${\nabla }_{q}F\left(z\right)=-{\displaystyle \frac{1}{q{z}^2}}+\sum _{k=1}^{\infty }{\left[k\right]}_q{a}_k{z}^{k-1},$$

(6)

where

$${\left[k\right]}_q={\displaystyle \frac{1-q^k}{1-q}}.$$

As $q\to 1^{-}$, ${\left[k\right]}_q=k$, and ${\nabla }_{q}F\left(z\right)=F^{\prime }\left(z\right)$.

In this paper, we introduce a new operator $D_q^n$ that serves as the basis for defining a new class of meromorphic functions, establishing sharp geometric bounds, and characterizing hydrodynamic stability.

Definition 4.

Using ${\nabla }_q$ and for $F\in \sum$, we define the operator $D_q^n:\sum \to \sum$ by

$$\begin{array}{cc}\hfill D_q^{0}F\left(z\right)& =F\left(z\right),\hfill \\ \hfill D_q^{1}F\left(z\right)& ={\displaystyle \frac{{\nabla }_q\left[z^{2}F\left(z\right)\right]}{z}}={\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{[k+2]}_q{a}_k{z}^k,\hfill \\ \hfill D_q^{2}F\left(z\right)& =D_q^1\left[D_q^{1}F\left(z\right)\right]={\displaystyle \frac{{\nabla }_q\left[z^2{D}_q^{1}F\left(z\right)\right]}{z}}={\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{[k+2]}_q^2{a}_k{z}^k\hfill \end{array}$$

and (in general)

$$D_q^{n}F\left(z\right)={\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{[k+2]}_q^n{a}_k{z}^k.$$

(7)

Motivated by previous works on meromorphic functions by function theorists (see [19,20,21,22,23,24]), we define the following new class of functions in ${\sum }_q^n\left(\alpha \right)$ by using the introduced operator $D_q^n$.

Definition 5.

The function $F\in {\sum }_q^n\left(\alpha \right)$ if it satisfies

$$\Re \left\{{\displaystyle \frac{D_q^{n+1}F\left(z\right)}{D_q^{n}F\left(z\right)}}-\alpha \right\}>\left|{\displaystyle \frac{D_q^{n+1}F\left(z\right)}{D_q^{n}F\left(z\right)}}-1\right|(n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}),$$

(8)

where $\mathbb{N}$ is the set of positive integers and for some α ($0\le \alpha <1$).

Note that

$$\underset{q\to 1^{-}}{lim}\sum _q^n\left(\alpha \right)=\sum ^n\left(\alpha \right)=\left\{F:\Re \left\{{\displaystyle \frac{D^{n+1}F\left(z\right)}{D^{n}F\left(z\right)}}-\alpha \right\}>\left|{\displaystyle \frac{D^{n+1}F\left(z\right)}{D^{n}F\left(z\right)}}-1\right|\right\}.$$

Also, we further introduce the subclass ${\sum }_q^n(\alpha ,c)\subset {\sum }_q^n\left(\alpha \right)$, which fixes the leading coefficient using the parameter c ($0\le c<1$).

Definition 6.

Let ${\sum }_q^n(\alpha ,c)\subset {\sum }_q^n\left(\alpha \right)$ consisting of

$$F\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}z+\sum _{k=2}^{\infty }{a}_k{z}^k,$$

(9)

with $0\le c<1$.

Following Goodman [12], Ruscheweyh [25], Altintas, and Owa [10], we define the $N_{q,\delta }$-neighborhood for functions in the class ∑.

Definition 7.

For $F\left(z\right)\in \sum$, the $N_{q,\delta }$-neighborhood is given by

$$N_{q,\delta }(F,g)=\left\{g:g\left(z\right)\in \sum ,g\left(z\right)={\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{b}_k{z}^k\mathit{and}\sum _{k=1}^{\infty }{\left[k\right]}_q|a_k-b_k|\le {\delta }_q\right\},$$

(10)

and for $e\left(z\right)={\displaystyle \frac{1}{z}}$;

$$N_{q,\delta }(e,g)=\left\{g:g\left(z\right)\in \sum ,g\left(z\right)={\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{b}_k{z}^k\mathit{and}\sum _{k=1}^{\infty }{\left[k\right]}_q\left|b_k\right|\le {\delta }_q\right\}.$$

(11)

3. Principal Findings

First, we introduce a necessary and sufficient condition that uses a sharp coefficient inequality to characterize the class ${\sum }_q^n\left(\alpha \right)$.

Theorem 1.

A function $F\in {\sum }_q^n\left(\alpha \right)$ if and only if

$$\sum _{k=1}^{\infty }(2{[k+2]}_q+\alpha -2){[k+2]}_q^n{a}_k\le 1-\alpha .$$

(12)

Proof. 

Let (12) be true. It is enough to prove that

$$\left|{\displaystyle \frac{D_q^{n+1}F\left(z\right)}{D_q^{n}F\left(z\right)}}-1\right|-\Re \left\{{\displaystyle \frac{D_q^{n+1}F\left(z\right)}{D_q^{n}F\left(z\right)}}-1\right\}\le 1-\alpha .$$

We have

$$\left|{\displaystyle \frac{D_q^{n+1}F\left(z\right)}{D_q^{n}F\left(z\right)}}-1\right|-\Re \left\{{\displaystyle \frac{D_q^{n+1}F\left(z\right)}{D_q^{n}F\left(z\right)}}-1\right\}\le 2\left|{\displaystyle \frac{D_q^{n+1}F\left(z\right)}{D_q^{n}F\left(z\right)}}-1\right|$$

$$\le {\displaystyle \frac{2{\sum }_{k=1}^{\infty }({[k+2]}_q-1){[k+2]}_q^n{a}_k}{1+{\sum }_{k=1}^{\infty }{[k+2]}_q^n{a}_k}}\le 1-\alpha .$$

Hence, $F\in {\sum }_q^n\left(\alpha \right)$. Conversely, let $F\in {\sum }_q^n\left(\alpha \right)$, then

$$\Re \left\{{\displaystyle \frac{1+{\sum }_{k=1}^{\infty }{[k+2]}_q^{n+1}{a}_k{z}^{k+1}}{1+{\sum }_{k=1}^{\infty }{[k+2]}_q^n{a}_k{z}^{k+1}}}-\alpha \right\}>\left|{\displaystyle \frac{{\sum }_{k=1}^{\infty }({[k+2]}_q-1){[k+2]}_q^n{a}_k{z}^{k+1}}{1+{\sum }_{k=1}^{\infty }{[k+2]}_q^n{a}_k{z}^{k+1}}}\right|.$$

(13)

Choosing z real and letting $z\to 1^{-}$, we get (12). □

The following corollary follows directly from the result of Theorem 1, which gives sharp coefficient estimates and validates the tightness of the bounds given in the theorem.

Corollary 1.

For $F\in {\sum }_q^n\left(\alpha \right)$, from (12), we have

$$a_k\le {\displaystyle \frac{1-\alpha }{(2{[k+2]}_q+\alpha -2){[k+2]}_q^n}}(k\ge 1).$$

(14)

The bound is sharp since it is attained by the function

$$F\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{1-\alpha }{(2{[k+2]}_q+\alpha -2){[k+2]}_q^n}}{z}^k,$$

(15)

that satisfies the coefficient condition given in (12) with only one non-zero coefficient $a_k(k\ge 1)$, and achieves the maximum possible value under the summation constraint. Note that any attempt to increase $a_k$ beyond this value would violate the defining condition of the class ${\sum }_q^n\left(\alpha \right)$, demonstrating that the bound cannot be improved. Furthermore, this extremal function is constructed by choosing the coefficients such that the inequality in Theorem 1 becomes an equality. Thus, the bound is the best possible, and the result is sharp.

Second, we introduce a parameterized subclass ${\sum }_q^n(\alpha ,c)$ to extend this result, and allow more accurate geometric control.

Theorem 2.

For $F\left(z\right)$ provided by (9), $F\left(z\right)\in {\sum }_q^n(\alpha ,c)$ if and only if

$$\sum _{k=2}^{\infty }(2{[k+2]}_q+\alpha -2){[k+2]}_q^n{a}_k\le (1-\alpha )(1-c).$$

(16)

Proof. 

Putting

$$a_1={\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}(0<c<1),$$

(17)

in (12), we have

$$c+\sum _{k=2}^{\infty }{\displaystyle \frac{(2{[k+2]}_q+\alpha -2){[k+2]}_q^n}{(1-\alpha )}}{a}_k\le 1,$$

(18)

which gives (16). The equality occurs for

$$F\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}z+{\displaystyle \frac{(1-\alpha )(1-c)}{(2{[k+2]}_q+\alpha -2){[k+2]}_q^n}}{z}^k(k\ge 2).$$

(19)

□

The following corollary follows directly from Equation (16), which gives sharp coefficient estimates and improves the outcome of Theorem 2.

Corollary 2.

If $F\left(z\right)\in {\sum }_q^n(\alpha ,c)$, then

$$a_k\le {\displaystyle \frac{(1-\alpha )(1-c)}{(2{[k+2]}_q+\alpha -2){[k+2]}_q^n}}(k\ge 2),$$

(20)

the equality occurs for $F\left(z\right)$ given by (19).

Now, we provide upper bounds for the sum of coefficients, further enhancing the structure of the subclass ${\sum }_q^n(\alpha ,c)$.

Theorem 3.

If $F\left(z\right)\in {\sum }_q^n(\alpha ,c)$, then

$$\begin{array}{cc}\hfill \sum _{k=2}^{\infty }{a}_k& \le {\displaystyle \frac{(1-\alpha )(1-c)}{(2{\left[4\right]}_q+\alpha -2){\left[4\right]}_q^n}},\hfill \end{array}$$

(21)

and

$$\begin{array}{cc}\hfill \sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k& \le {\displaystyle \frac{{\left[2\right]}_q(1-\alpha )(1-c)}{(2{\left[4\right]}_q+\alpha -2){\left[4\right]}_q^n}}.\hfill \end{array}$$

(22)

Both bounds are sharp, achieved by

$$F\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}z+{\displaystyle \frac{(1-\alpha )(1-c)}{(2{\left[4\right]}_q+\alpha -2){\left[4\right]}_q^n}}{z}^2.$$

Proof. 

Let $F\left(z\right)\in {\sum }_q^n(\alpha ,c)$. Then, from (16), we have

$$(2{\left[4\right]}_q+\alpha -2){\left[4\right]}_q^n\sum _{k=2}^{\infty }{a}_k\le (1-\alpha )(1-c),$$

(23)

which gives (21). From (23) and (16), we have

$${\displaystyle \frac{(2{\left[4\right]}_q+\alpha -2){\left[4\right]}_q^n}{{\left[2\right]}_q}}\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k\le \sum _{k=2}^{\infty }(2{[k+2]}_q+\alpha -2){[k+2]}_q^m{a}_k$$

$$\le (1-\alpha )(1-c),$$

(24)

which gives (22). □

In addition, we establish distortion bounds for functions in the subclass ${\sum }_q^n(\alpha ,c)$.

Theorem 4.

For $F\left(z\right)\in {\sum }_q^n(\alpha ,c)$ and $0<\left|z\right|=r<1$, we get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{r}}-{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}r-{\displaystyle \frac{(1-\alpha )(1-c)}{(2{\left[4\right]}_q+\alpha -2){\left[4\right]}_q^n}}{r}^2\le \left|F\left(z\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{r}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}r+{\displaystyle \frac{(1-\alpha )(1-c)}{(2{\left[4\right]}_q+\alpha -2){\left[4\right]}_q^n}}{r}^2,\hfill \end{array}$$

(25)

with equality for

$$F\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}z+{\displaystyle \frac{(1-\alpha )(1-c)}{(2{\left[4\right]}_q+\alpha -2){\left[4\right]}_q^n}}{z}^2,$$

with $0\le c<1$.

Proof. 

For $F\left(z\right)\in {\sum }_q^n(\alpha ,c)$, we have

$$\left|F\left(z\right)\right|=\left|{\displaystyle \frac{1}{z}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}z+\sum _{k=2}^{\infty }{a}_k{z}^k\right|$$

$$\le {\displaystyle \frac{1}{\left|z\right|}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}\left|z\right|+{\left|z\right|}^2\sum _{k=2}^{\infty }{a}_k$$

$$\le {\displaystyle \frac{1}{r}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}r+r^2\sum _{k=2}^{\infty }{a}_k,$$

and

$$\left|F\left(z\right)\right|=\left|{\displaystyle \frac{1}{z}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}z+\sum _{k=2}^{\infty }{a}_k{z}^k\right|$$

$$\ge {\displaystyle \frac{1}{\left|z\right|}}-{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}\left|z\right|-{\left|z\right|}^2\sum _{k=2}^{\infty }{a}_k$$

$$\ge {\displaystyle \frac{1}{r}}-{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}r-r^2\sum _{k=2}^{\infty }{a}_k,$$

from (21), we have (25). □

Furthermore, we establish distortion bounds for the q-derivative of functions in the subclass ${\sum }_q^n(\alpha ,c)$, enhancing the understanding of the functions’ differential behavior.

Theorem 5.

For $F\left(z\right)\in {\sum }_q^n(\alpha ,c)$ and $0<\left|z\right|=r<1$, we get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{q{r}^2}}-{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}-{\displaystyle \frac{{\left[2\right]}_q(1-\alpha )(1-c)}{(2{\left[4\right]}_q+\alpha -2){\left[4\right]}_q^n}}r\le \left|{\nabla }_{q}F\left(z\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{q{r}^2}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}+{\displaystyle \frac{{\left[2\right]}_q(1-\alpha )(1-c)}{(2{\left[4\right]}_q+\alpha -2){\left[4\right]}_q^n}}r,\hfill \end{array}$$

(26)

with equality for

$${\nabla }_{q}F\left(z\right)={\displaystyle \frac{1}{q{z}^2}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}+{\displaystyle \frac{{\left[2\right]}_q(1-\alpha )(1-c)}{(2{\left[4\right]}_q+\alpha -2){\left[4\right]}_q^n}}z.$$

Proof. 

For $F\left(z\right)\in {\sum }_q^n(\alpha ,c)$, we have

$$\begin{array}{cc}& |{\nabla }_{q}F\left(z\right)|=\left|-{\displaystyle \frac{1}{q{z}^2}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}+\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k{z}^{k-1}\right|\hfill \\ & \le {\displaystyle \frac{1}{{q\left|z\right|}^2}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}+\left|z\right|\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k\hfill \\ & \le {\displaystyle \frac{1}{q{r}^2}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}+r\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k,\hfill \end{array}$$

and

$$\begin{array}{cc}& |{\nabla }_{q}F\left(z\right)|=\left|-{\displaystyle \frac{1}{q{z}^2}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}+\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k{z}^{k-1}\right|\hfill \\ & \ge {\displaystyle \frac{1}{{q\left|z\right|}^2}}-{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}-\left|z\right|\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k\hfill \\ & \ge {\displaystyle \frac{1}{q{r}^2}}-{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}-r\sum _{k=2}^{\infty }{\left[k\right]}_q{a}_k,\hfill \end{array}$$

from (21), we have (26). □

Next, we determine the radius of starlikeness for the functions in ${\sum }_q^n(\alpha ,c)$, identifying the maximal region where the starlike property holds.

Theorem 6.

Let $F\left(z\right)\in {\sum }_q^n(\alpha ,c)$. Then $F\left(z\right)$ is q-starlike of order v ($0\le v<1$) in $\left|z\right|<r_1$, where $r_1$ is the highest possible value for which

$${\displaystyle \frac{(3-v)(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}{r}^2+{\displaystyle \frac{(1-\alpha )(k_0-v+2)(1-c)}{(2{[k_0+2]}_q+\alpha -2){[k_0+2]}_q^n}}{r}^{k+1}\le 1-v(k\ge 2).$$

(27)

The sharpness is provided by (19).

Proof. 

It suffices to demonstrate that

$$\left|{\displaystyle \frac{z\left({\nabla }_{q}F\left(z\right)\right)}{F\left(z\right)}}+1\right|\le 1-v\left(\right|z|<r_1).$$

(28)

Be aware that

$$\left|{\displaystyle \frac{z\left({\nabla }_{q}F\left(z\right)\right)}{F\left(z\right)}}+1\right|\le {\displaystyle \frac{\frac{2(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}\left|z\right|+{\sum }_{k=2}^{\infty }({\left[k\right]}_q+1)a_k{\left|z\right|}^k}{\frac{1}{\left|z\right|}-\frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}\left|z\right|-{\sum }_{k=2}^{\infty }{a}_k{\left|z\right|}^k}}$$

$$\le 1-v,$$

(29)

for $\left|z\right|<r_1$, this holds if

$${\displaystyle \frac{(3-v)(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}{r}_1^2+\sum _{k=2}^{\infty }({\left[k\right]}_q-v+2)a_k{r}_1^{k+1}\le 1-v.$$

From (16), we may take

$$a_k\le {\displaystyle \frac{(1-\alpha )(1-c){\lambda }_k}{(2{[k+2]}_q+\alpha -2){[k+2]}_q^n}}(k\ge 2),$$

${\lambda }_k\ge 0$ and ${\sum }_{k=2}^{\infty }{\lambda }_k\le 1$. For every fixed r, we choose the positive integer $k_0=k_0\left(r\right)$ for which

$${\displaystyle \frac{(1-\alpha )({\left[k_0\right]}_q-v+2)(1-c)}{(2{[k_0+2]}_q+\alpha -2){[k_0+2]}_q^n}}{r}^{k_0+1},\mathrm{is}\mathrm{maximal}.$$

It follows that

$$\sum _{k=2}^{\infty }({\left[k\right]}_q-v+2)a_k{r}^{k+1}\le {\displaystyle \frac{(1-\alpha )({\left[k_0\right]}_q-v+2)(1-c)}{(2{[k_0+2]}_q+\alpha -2){[k_0+2]}_q^n}}{r}^{k_0+1},$$

then F is starlike of order v in $\left|z\right|<r_1$, provided that

$${\displaystyle \frac{(3-v)(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}{r}_1^2+{\displaystyle \frac{(1-\alpha )({\left[k_0\right]}_q-v+2)(1-c)}{(2{[k_0+2]}_q+\alpha -2){[k_0+2]}_q^n}}{r}_1^{k_0+1}\le 1-v.$$

We find the value $r_1=r_0(\alpha ,c,v,k)$ and the related integer $k_0\left(r_0\right)$ such that

$${\displaystyle \frac{(3-v)(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}{r}_0^2+{\displaystyle \frac{(1-\alpha )({\left[k_0\right]}_q-v+2)(1-c)}{(2{[k_0+2]}_q+\alpha -2){[k_0+2]}_q^n}}{r}_0^{k_0+1}=1-v.$$

Then $r_0$ is the radius of starlikeness of order v for $F\in {\sum }_q^n(\alpha ,c)$. To verify sharpness, from the equality function given in (19), the left-hand side of the condition given in (29) reaches the value $1-v$ at $z=r_1$ with $r_1$ real and positive, meaning that for any larger radius $r>r_1$, the inequality fails. Hence, $r_1$ is the largest possible radius within which all functions in the class ${\sum }_q^n(\alpha ,c)$ remain q-starlike of order v, and no larger radius can be found. This proves that the result is sharp, and the bound cannot be improved. □

In addition, we prove that the subclass ${\sum }_q^n(\alpha ,c)$ is closed under convex linear combinations, which guarantees stability under such operations.

Theorem 7.

Under convex linear combinations, ${\sum }_q^n(\alpha ,c)$ is closed.

Proof. 

Let F be defined by (9) and

$$h\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}z+\sum _{k=2}^{\infty }{b}_k{z}^k(b_k\ge 0).$$

(30)

Letting $F,h\in {\sum }_q^n(\alpha ,c)$, we must demonstrate that $G\in {\sum }_q^n(\alpha ,c)$, where

$$G\left(z\right)=\zeta F\left(z\right)+(1-\zeta )h\left(z\right)(0\le \zeta \le 1).$$

(31)

Since

$$G\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}z+\sum _{k=2}^{\infty }\{\zeta a_k+(1-\zeta )b_k\}{z}^k,$$

(32)

then

$$\sum _{k=2}^{\infty }(2{[k+2]}_q+\alpha -2){[k+2]}_q^n\{\zeta a_k+(1-\zeta )b_k\}\le (1-\alpha )(1-c).$$

(33)

From (16), $G\left(z\right)\in {\sum }_q^n(\alpha ,c)$. □

Next, we show that each function in the class can be expressed as a convex combination of extreme points, providing a constructive representation.

Theorem 8.

Let

$$F_1\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}z,$$

(34)

and

$$F_k\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}z+{\displaystyle \frac{(1-\alpha )(1-c)}{(2{[k+2]}_q+\alpha -2){[k+2]}_q^n}}{z}^k,$$

(35)

for $k\ge 2$. Then $F\in {\sum }_q^n(\alpha ,c)$ if and only if

$$F\left(z\right)=\sum _{k=2}^{\infty }{\eta }_k{F}_k\left(z\right),$$

(36)

where ${\eta }_k\ge 0$, and

$$\sum _{k=2}^{\infty }{\eta }_k\le 1.$$

(37)

Proof. 

Let $F\left(z\right)$ be given by (36). Then from (34), (35), and (37), we get

$$F\left(z\right)={\displaystyle \frac{1}{z}}+{\displaystyle \frac{(1-\alpha )c}{(2{\left[3\right]}_q+\alpha -2){\left[3\right]}_q^n}}z+\sum _{k=2}^{\infty }{\displaystyle \frac{(1-\alpha )(1-c){\eta }_k}{(2{[k+2]}_q+\alpha -2){[k+2]}_q^n}}{z}^k.$$

(38)

Since

$$\sum _{k=2}^{\infty }{\displaystyle \frac{(1-\alpha )(1-c){\eta }_k}{(2{[k+2]}_q+\alpha -2){[k+2]}_q^n}}·{\displaystyle \frac{(2{[k+2]}_q+\alpha -2){[k+2]}_q^n}{(1-\alpha )(1-c)}}$$

$$=\sum _{k=2}^{\infty }{\eta }_k=1-{\eta }_1\le 1,$$

(39)

then, from Theorem 2, $F\in {\sum }_q^n(\alpha ,c)$. Conversely, let $F\in {\sum }_q^n(\alpha ,c)$, satisfy (20) for $k\ge 2$, then

$${\eta }_k={\displaystyle \frac{(2{[k+2]}_q+\alpha -2){[k+2]}_q^n}{(1-\alpha )(1-c)}}{a}_k\le 1,$$

(40)

and

$${\eta }_1=1-\sum _{k=2}^{\infty }{\eta }_k.$$

(41)

□

Corollary 3.

The extreme points of ${\sum }_q^n(\alpha ,c)$ are $F_k\left(z\right)$ ($k\ge 2$) provided by (34) and (35).

Finally, we investigate the neighborhood structure of ${\sum }_q^n(\alpha ,c)$, giving bounds that describe how functions behave under small perturbations.

Theorem 9.

If $F\in {\sum }_q^n(\alpha ,c)$, then

$$\sum _q^n(\alpha ,c)\subset N_{\xi }(F,q),$$

(42)

where ξ is given by

$$\xi ={\displaystyle \frac{{\left[2\right]}_q(1-\alpha )(1-c)}{(2{\left[4\right]}_q+\alpha -2){\left[4\right]}_q^n}}.$$

(43)

Proof. 

For $F\in {\sum }_q^n(\alpha ,c)$ from (22) and (11), we get (43). □

4. Fluid Mechanics Applications

Fluid mechanics is the area of physics that studies how forces affect the motion of gases and liquids. Fluid mechanics has implications in everything from chemical and mechanical engineering to geophysics and astrophysics because of its numerous impact interactions. This section presents some problems within the realm of fluid mechanics, serving as examples to examine the solutions of two-dimensional (2D) potential flow issues across a circular cylinder by both vortex and source/sink methods.

A flow is characterized as 2D when the velocity is confined to a fixed plane and does not depend on the coordinate perpendicular to that plane. A source flow comprises a symmetrical flow field characterized by radial streamlines and a line source from which the fluid emanates and traverses a plane perpendicular to the line [26,27]. The area of flow expands as the fluid moves outward. It is accurate to say that when the continuity equation is satisfied, the streamlines spread out and the velocity drops. Therefore, at a given radial distance, the velocity values are the same at every point.

We concentrate on 2D potential flow, which is defined as irrotational, inviscid, and incompressible. For such flows, the complex potential is represented by $W\left(z\right)=\phi (x,y)+i\psi (x,y)$, where $\phi$ is the velocity potential, and $\psi$ is the stream function and an analytic function, and excludes isolated singularities that correspond to flow features such as vortices or sources in the flow domain. The velocity potential $\phi$ and the stream function $\psi$ are solutions to Laplace’s equation, ${\nabla }^2\phi =0$, ${\nabla }^2\psi =0$ and describe such flows. The meromorphic operator $D_q^n$ is defined for functions belonging to the class ∑. Although the complex potential $W\left(z\right)$ for sources or vortices often involves $logz$, which is not in ∑, its derivative, the complex velocity $W^{\prime }\left(z\right)$, can be a function of ∑. We will explore how the operator $D_q^n$ and the properties of associated function classes ${\sum }_q^n\left(\alpha \right)$ and ${\sum }_q^n(\alpha ,c)$ can be applied to analyze such flows, particularly those involving sources/sinks and vortices in punctured domains.

4.1. Application 1: Source/Sink in 2D Potential Flow

Let K be the strength flow rate. A source ($K>0$) or sink ($K<0$) at a point $z_0$ produces radial lines known as streamlines that emerge from or converge to $z_0$. The complex potential at $z_0=0$ is the following:

$$W_1\left(z\right)={\displaystyle \frac{K}{2\pi }}logz={\displaystyle \frac{K}{2\pi }}(ln|z|+iargz).$$

Here,

$$\phi ={\displaystyle \frac{K}{2\pi }}ln\left|z\right|\mathrm{and}\psi ={\displaystyle \frac{K}{2\pi }}argz.$$

The complex velocity $W_1^{\prime }\left(z\right)={\displaystyle \frac{K}{2\pi z}}$ is meromorphic and belongs to ${\sum }_q^n\left(\alpha \right)$ with $a_k=0$ for $k\ge 1$.

• Operator Application and Distortion Bounds:
  Applying the operator $D_q^n$ to the complex velocity $W_1^{\prime }\left(z\right)$:

  $$D_q^n\left(W_1^{\prime }\left(z\right)\right)=D_q^n\left({\displaystyle \frac{K}{2\pi z}}\right)={\displaystyle \frac{K}{2\pi }}\left({\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{[k+2]}_q^2·0·z^k\right)={\displaystyle \frac{K}{2\pi z}},$$

  i.e., $D_q^n\left(W_1^{\prime }\left(z\right)\right)=W_1^{\prime }\left(z\right)$, this shows that the operator $D_q^n$ maintains the complex velocity of a pure source/sink flow. Then from Theorem 4, we have

  $$|W_1^{\prime }\left(z\right)|\le {\displaystyle \frac{\left|K\right|}{2\pi r}}(0<|z|=r<1),$$

  which validates that the velocity field is exactly represented without distortion.
  To confirm the invariance of the operator, we visualize a source flow when $K=2$ on the punctured disk $0.1<\left|z\right|<0.9$. Figure 1 confirms that $D_q^n$ preserves radial streamlines with velocity decaying as ${\displaystyle \frac{1}{r}}$, where the parameter K controls flow strength.
• Starlikeness and Flow Geometry:
  By Theorem 6, we have

  $$\left|{\displaystyle \frac{z{W}^{\prime \prime }\left(z\right)}{W^{\prime }\left(z\right)}}+1\right|=0<1-v(0\le v<1)$$

  and this implies that the flow is starlike in $U^{*}$ for all v, meaning that the flow is purely radial (no tangential component) and has no stagnation points.
• Neighborhood Stability for Source/Sink Flow:
  Theorem 9 gives the neighborhood

  $$N_{q,\delta }(W_1^{\prime }\left(z\right),g)\subset N_{q,\xi }(W_1^{\prime }\left(z\right),g),$$

  where $\xi$ is given by (43), which defines crucial thresholds for source/sink flows to preserve structural integrity in the face of perturbations. When the velocity field $g\left(z\right)$ is disturbed and satisfies

  $$\sum _{k=1}^{\infty }{\left[k\right]}_q\left|b_k\right|\le \xi ,$$

  which maintains the following:
  • Consistency of the radial pattern: Strictly radial streamlines.
  • Prevention of zero-velocity points: No stagnation points appear in $U^{*}$.
  The coefficients $b_k$ define the perturbation. We employ a particular perturbation:

  $$g_1\left(z\right)=W_1^{\prime }\left(z\right)+\epsilon z^2.$$

  This means that there is just one non-zero coefficient $b_2$ and $b_k=0$ for $k\ne 2$. Let $\epsilon =|b_2|$, then the neighborhood condition becomes

  $$\sum _{k=1}^{\infty }{\left[k\right]}_q\left|b_k\right|={\left[2\right]}_q\epsilon \le \xi .$$

  Taking $q=0.7$, $\alpha =0.2$, $c=0.4$, $n=0$ and from (43), we get $\xi =0.3$, and we can compute

  $${\left[2\right]}_q={\displaystyle \frac{1-q^2}{1-q}}=1.7.$$

  Then for $\epsilon =0.06$ (stable), we have

  $$\sum _{k=1}^{\infty }{\left[k\right]}_q\left|b_k\right|=0.1\le \xi =0.3,$$

  as shown in Figure 2.
  Now, for $\epsilon =0.45$ (unstable), we have

  $$\sum _{k=1}^{\infty }{\left[k\right]}_q\left|b_k\right|=0.8>\xi =0.3,$$

  which destroys radial symmetry and creates stagnation points as shown in Figure 3.

4.2. Application 2: Source/Sink with Vortex at Same Point

As is often known, a vortex is made up of a point where a fluid flows in circles [27]. By switching the velocity potential and the stream function for a source or sink, the appropriate complex potential can be obtained. A vortex (circulation strength L) has concentric streamlines. The complex potential function for a vortex of circulation strength L is

$$W\left(z\right)={\displaystyle \frac{iL}{2\pi }}logz,$$

while the complex potential function in the case of a source with a vortex at $z_0=0$ is

$$W_2\left(z\right)={\displaystyle \frac{K+iL}{2\pi }}logz,$$

then

$$W_2^{\prime }\left(z\right)={\displaystyle \frac{K+iL}{2\pi z}},$$

which is meromorphic and lies in ${\sum }_q^n\left(\alpha \right)$.

• Operator Application and Distortion Bounds:
  Applying the operator $D_q^n$ to the complex velocity $W_2^{\prime }\left(z\right)$:

  $$D_q^n\left(W_2^{\prime }\left(z\right)\right)=D_q^n\left({\displaystyle \frac{K+iL}{2\pi z}}\right)={\displaystyle \frac{K+iL}{2\pi }}\left({\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{[k+2]}_q^2·0·z^k\right)={\displaystyle \frac{K+iL}{2\pi z}},$$

  i.e., $D_q^n\left(W_2^{\prime }\left(z\right)\right)=W_2^{\prime }\left(z\right)$, this shows that the operator $D_q^n$ maintains the combined source-vortex flow. Then from Theorem 4 we have

  $$|W_2^{\prime }\left(z\right)|\le {\displaystyle \frac{\sqrt{K^2+L^2}}{2\pi r}}(0<|z|=r<1),$$

  which gives the exact bound for the unperturbed case.
  Taking $K=1$, $L=0.5$, and $q=0.8$ to validate operator invariance, Figure 4 compares the original vortex–source flow with its image under the operator $D_q^n$. The identical streamline structure verifies that $D_q^n$ maintains the complex velocity field and the distinctive logarithmic spirals without stagnation sites or distortion.
• q-Derivative Bounds for complex velocity:
  By Theorem 5, we have

  $$|{\nabla }_q\left(W_2^{\prime }\left(z\right)\right)|\le {\displaystyle \frac{\sqrt{K^2+L^2}}{2\pi q{r}^2}},$$

  whcih quantifies how sharply the velocity field varies near the singularity ($z=0$).
• Stagnation Points:
  By Theorem 6, we have

  $$\left|{\displaystyle \frac{z{W}_2^{\prime \prime }\left(z\right)}{W_2^{\prime }\left(z\right)}}+1\right|=0\le 1-v,$$

  and therefore, the flow has no stagnation points, which means that its streamlines are logarithmic spirals.
• Neighborhood Stability for Vortex Source Flows:
  Theorem 9 gives the neighborhood

  $$N_{q,\delta }(W_2^{\prime }\left(z\right),g)\subset N_{q,\xi }(W_2^{\prime }\left(z\right),g),$$

  where $\xi$ is given by (43), and this bounds perturbations in flow velocity,

  $$\sum _{k=1}^{\infty }{\left[k\right]}_q\left|b_k\right|\le \xi ,$$

  ensuring the perturbed flow retains the spiral streamline structure without breaking into closed loops and stagnation points.
  To numerically validate our stability bounds, we choose the following physically realistic parameters:
  Flow strengths $K=1$ and $L=0.5$: Ensures comparable source/vortex dominance while avoiding degeneracy ($K>L$ preserves radial-to-spiral transition).
  Operator parameter $q=0.7$: Exhibits distinct q-derivative effects without asymptotic behavior.
  Critical radius $\left|z\right|=0.5$: Positions analysis midway between singularity ($z=0$) and boundary ($\left|z\right|=1$), where perturbations are measurable.
  These choices optimize the visualization of stagnation thresholds under controlled disturbances. Figure 5 quantifies stability bounds for vortex–source flow. We calculate the magnitudes of velocity under three different perturbation energies ${\sum }_{k=1}^{\infty }{\left[k\right]}_q\left|b_k\right|=\{0.1,0.25,0.4\}$, contrasting with the threshold $\xi =0.3$. Then

  $$|W_2^{\prime }\left(z\right)|\le {\displaystyle \frac{\sqrt{K^2+L^2}}{2\pi r}}=0.356.$$
  Compliant perturbations stay below $\xi$, while violations cause velocity spikes exceeding the 300% critical threshold.
  To visualize the physical consequences of stability bounds at $K=1$, $L=0.5$, and $\xi =0.3$, we employ a particular perturbation:

  $$g_2\left(z\right)=W_2^{\prime }\left(z\right)+\epsilon z^2,$$

  where $\epsilon$ is the quadrupole perturbation coefficient, under two perturbation scenarios:
  • Stable case: ${\sum }_{k=1}^{\infty }{\left[k\right]}_q\left|b_k\right|=0.25\le \xi =0.3\Rightarrow \epsilon =0.147$.
  • Unstable case: ${\sum }_{k=1}^{\infty }{\left[k\right]}_q\left|b_k\right|=0.4>\xi =0.3\Rightarrow \epsilon =0.235$.
  Using stream function color maps, streamline tracings, and stagnation point detection, we demonstrate how compliant perturbations preserve spiral connectivity while violations cause geometrical fragmentation (Figure 6).

5. Conclusions

Through this study, meromorphic q-operators are established as transformative tools for both mathematical innovation and fluid mechanics. Theoretically, we develop geometric function theory by establishing sharp boundaries for a new meromorphic function class, including coefficient estimates, distortion theorems, starlikeness, convex combination closure, extreme point characterization, and we study the simple notion of the neighborhood. Practically, we present applications in hydrodynamics: identifying stagnation points in vortex flows, predicting velocity decline in source/sink systems, and determining stability thresholds that protect crucial streamlines from perturbations. These developments give mathematics a solid foundation for singularity analysis and engineers unprecedented control over atmospheric models and microscale fluidic systems. Future work will apply this operator-based approach to 3D turbulent flows and magnetohydrodynamics, where meromorphic univalence holds promise for novel understandings of the creation and management of singularities.