Generalization of the Rafid Operator and Its Symmetric Role in Meromorphic Function Theory with Electrostatic Applications

Abstract

This study introduces a new integral operator $I_{p,\mu }^{\delta }$ that extends the traditional Rafid operator to meromorphic p-valent functions. Using this operator, we define and investigate two new subclasses: ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right)$, consisting of functions with nonnegative coefficients, and ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$, which further fixes the second positive coefficient. For these classes, we establish a necessary and sufficient coefficient condition, which serves as the foundation for deriving a set of sharp results. These include accurate coefficient bounds, distortion theorems for functions and derivatives, and radii of starlikeness and convexity of a specific order. Furthermore, we demonstrate the closure property of the class ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$, identify its extreme points, and then construct a neighborhood theorem. All the findings presented in this paper are sharp. To demonstrate the practical utility of our symmetric operator paradigm, we apply it to a canonical fractional electrodynamics problem. We demonstrate how sharp distortion theorems establish rigorous, time-invariant upper bounds for a solitary electrostatic potential and its accompanying electric field, resulting in a mathematically guaranteed safety buffer against dielectric breakdown. This study develops a symmetric and consistent approach to investigating the geometric characteristics of meromorphic multivalent functions and their applications in physical models.

1. Introduction

The geometric theory of meromorphic functions dates back to Karl Weierstrass [1], who established their singularity structure through partial fraction decompositions. Mittag-Leffler [2] expanded this by developing canonical representations for functions with infinite poles. The field was transformed when Pommerenke [3] introduced meromorphically starlike functions, described by the geometric condition $-\Re \left\{z{f}^{\prime }/f\right\}>0.$ This class, inspired by Alexander’s [4] starlike univalent functions, was critical for mapping radius difficulties. Miller [5] expanded this concept to meromorphically convex functions with $-\Re \left\{1+z{f}^{″}/f^{\prime }\right\}>0$, relating complex derivatives to geometric distortion.

Kumar and Shukla [6] pioneered the generalization to p-valent meromorphic functions, which have poles of order p. Their approach enabled the modeling of higher-order singularities (${\left|z\right|}^{-p},$ $p>1$). This approach revealed complex relationships between valency, coefficient bounds, and geometric features, which were later developed by many authors [7,8,9].

Operators have made significant contributions to the advancement of geometric function theory. Ruscheweyh [10] developed convolution operators for univalent functions, while Altıntaş and Srivastava [11] created the first operators that preserve meromorphic starlikeness. Recent developments include Aouf’s q-differential operators for distortion control [9] and Totoi’s subordination-preserving integrals [12]; see also [13]. A particularly influential operator in the study of analytic univalent functions is the Rafid operator [14]. Its success in producing new classes and facilitating the identification of sharp geometric characteristics motivates its wider application.

In this study, we bridge a significant gap in the literature by generalizing the Rafid operator to the class of meromorphic p-valent functions. We introduce the new operator, $I_{p,\mu }^{\delta }$, which is a powerful tool for studying this family of functions. Using this operator, we introduce two new subclasses ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right)$ and ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$. Also, we provide a full set of sharp results for these classes, including coefficient bounds, distortion theorems, radii of starlikeness and convexity, closure under convex linear combinations, extreme points, and neighborhood results. As a result, this work methodically integrates and expands on classical meromorphic function theory results, laying a solid platform for future research. To demonstrate the practical utility of this unified approach, we devote a section to its application in obtaining rigorous safety bounds for solitary electrostatic potentials governed by fractional diffusion equations.

The manuscript flows naturally: Section 2 contains the core definitions, notation, and mathematical preliminaries. Section 3 advances the theoretical results. Section 4 shows how our theory may be applied to a fractional diffusion model in electrodynamics to obtain rigorous bounds for singular potential fields. Section 5 provides concluding observations and suggestions for future study.

2. Preliminaries and Definitions

The mathematical foundation for this work is based on the class ${\Sigma }_p$ of analytic and p-valent functions defined on the punctured open unit disk $U^{*}=\{z\in \mathbb{C}:0<|z|<1\}$. Every function $f\in {\Sigma }_p$ has a Laurent series expansion of the form

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

(1)

$p\in \mathbb{N}=\{1,2,\dots \}$. Our analysis will focus on a specific subclass of ${\Sigma }_p$, consisting of functions with non-negative coefficients. A function $f\left(z\right)$ belongs to ${\Sigma }_p^{+}$ if it has the form

$$f\left(z\right)={\displaystyle \frac{1}{z^p}}+\sum _{k=1}^{\infty }{a}_k{z}^k,(a_k\ge 0).$$

(2)

Definition 1.

Let two functions $f_j\left(z\right)$$\in {\Sigma }_p$, $(j=1,2)$ be defined by

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

and the Hadamard product (or convolution) of $f_1\left(z\right)$ and $f_2\left(z\right)$, denoted by $(f_1\ast f_2)\left(z\right)$, is defined as

$$\left(f_1\ast f_2\right)z={\displaystyle \frac{1}{z^p}}+\sum _{k=1}^{\infty }{a}_{k,1}{a}_{k,2}{z}^k.$$

The Hadamard product is commutative by definition, which means that $(f_1\ast f_2)\left(z\right)=(f_2\ast f_1)\left(z\right)$. Additionally, if $f_1,f_2\in {\Sigma }_p^{+}$, then $(f_1\ast f_2)\in {\Sigma }_p^{+}$.

Definition 2.

Pommerenke [3] established the foundational geometric classes for univalent meromorphic functions ($p=1$). He introduced the class of meromorphically starlike functions, ${\Sigma }^{*}$, as functions $f\in {\Sigma }_1$ satisfying

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

Shortly after, Miller [5] defined the analogous class of meromorphically convex functions, ${\Sigma }^c$, satisfying

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

These definitions were later refined to include an order $\alpha \in [0,1)$, leading to the classes ${\Sigma }^{*}\left(\alpha \right)$ and ${\Sigma }^c\left(\alpha \right)$. Kumar and Shukla [6] extended the definitions to the class ${\Sigma }_p$. The framework defines a function $f\in {\Sigma }_p$ as p-valent meromorphically starlike of order α if

$$-\Re \left({\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right)>\alpha ,(0\le \alpha <p),$$

and p-valent meromorphically convex of order α if

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

These p-valent classes form the geometric foundation for the new subclasses we introduce in this paper.

The employment of linear operators is an effective way to define and investigate new subclasses of analytic and meromorphic functions. The Rafid operator, originally defined for analytic univalent functions, has proved useful in the study of geometric features [14]. This operator and its variants have been widely researched for several classes of functions [15,16,17,18,19,20]; nevertheless, a generalization to the class of p-valent meromorphic functions remains unexplored. Motivated by this gap and the established efficacy of operator-based techniques, we develop a new generalized integral operator, inspired by the Rafid operator, specially built for the class ${\Sigma }_p$.

Definition 3.

For $p\in \mathbb{N}$, $0\le \delta \le 1$, and $0\le \mu <1,$ we introduce the operator $I_{p,\mu }^{\delta }:{\Sigma }_p\to {\Sigma }_p$ as

$$I_{p,\mu }^{\delta }f\left(z\right)={\displaystyle \frac{1}{{\left(p-\mu \right)}^{\delta +1}\mathrm{\Gamma }\left(\delta +1\right)}}\underset{0}{\overset{\infty }{\int }}{t}^{p+\delta }{e}^{-\left({\displaystyle \frac{t}{p-\mu }}\right)}f\left(tz\right)dt.$$

This integral is well-defined for $f\in {\Sigma }_p$ and $z\in U^{*}$ due to the exponential decay of the kernel, ensuring the function $t\mapsto t^{p+\delta }{e}^{-t/(p-\mu )}f\left(tz\right)$ over $[0,\infty )$. By substituting the Laurent series (1) of f into the integral and following Fubini’s Theorem (supported by the non-negativity of the integrand for $t\ge 0$ and the absolute convergence of the resulting series), we can obtain the corresponding series representation:

$$I_{p,\mu }^{\delta }f\left(z\right)={\displaystyle \frac{1}{z^p}}+\sum _{k=1}^{\infty }\phi \left(k,\delta ,\mu \right)a_k{z}^k,$$

(3)

where the kernel function $\phi \left(k,\delta ,\mu \right)$ is provided by

$$\phi \left(k,\delta ,\mu \right)={\displaystyle \frac{{\left(p-\mu \right)}^{k+p}\mathrm{\Gamma }\left(k+p+\delta +1\right)}{\mathrm{\Gamma }\left(\delta +1\right)}}.$$

(4)

In addition, the operator obeys the following differential relation:

$$z{\left(I_{p,\mu }^{\delta }f\left(z\right)\right)}^{\prime }=\left(\delta +1\right)I_{p,\mu }^{\delta +1}f\left(z\right)-\left(p+\delta +1\right)I_{p,\mu }^{\delta }f\left(z\right).$$

Note that

1. 

$I_{p,\mu }^{0}f\left(z\right)=L_p^{\mu }f\left(z\right)={\displaystyle \frac{1}{z^p}}+{\sum }_{k=1}^{\infty }{\left(p-\mu \right)}^{k+p}\mathrm{\Gamma }\left(k+p+1\right)a_k{z}^k;$

2. 

$I_{p,\mu }^{1}f\left(z\right)=M_p^{\mu }f\left(z\right)={\displaystyle \frac{1}{z^p}}+{\sum }_{k=1}^{\infty }{\left(p-\mu \right)}^{k+p}\mathrm{\Gamma }\left(k+p+2\right)a_k{z}^k.$

Building on the operator $I_{p,\mu }^{\delta }$ defined above, we will now introduce a new subclass of p-valent meromorphic functions.

Definition 4.

We introduce the subclass ${\Sigma }_p(\delta ,\mu ,\alpha )$ as the set of all $f\in {\Sigma }_p$ for which the following holds:

$$\left|{\displaystyle \frac{z{\left(I_{p,\mu }^{\delta }f\left(z\right)\right)}^{\prime }}{I_{p,\mu }^{\delta }f\left(z\right)}}+p\right|<\left|{\displaystyle \frac{z{\left(I_{p,\mu }^{\delta }f\left(z\right)\right)}^{\prime }}{I_{p,\mu }^{\delta }f\left(z\right)}}+2\alpha -p\right|,$$

(5)

which is geometrically equivalent to the standard condition for meromorphic starlikeness of order α $\left(0\le \alpha <p\right).$ Much of our subsequent analysis focuses on the subclass of non-negative coefficient functions. We denote this subclass by ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right)$, which is formally defined as the intersection

$${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right)={\Sigma }_p\left(\delta ,\mu ,\alpha \right)\cap {\Sigma }_p^{+},$$

where ${\Sigma }_p^{+}$ is the class of functions with non-negative coefficients. This subclass is particularly important since the positive constraint on the coefficients often allows the derivation of necessary and sufficient conditions, which is a prominent subject in our primary findings.

The parameters for the class ${\Sigma }_p\left(\delta ,\mu ,\alpha \right)$ and its subclasses fall between the following ranges:

$$p\in \mathbb{N},0\le \delta \le 1,0\le \mu <1,0\le \alpha <p.$$

The limiting behavior of these parameters relates our generalized operator to well-known traditional operators and classes. For example,

(i) The limitations $\delta \to 0^{+}$ and $\delta \to 1^{-}$, respectively, reduce the operator $I_{p,\mu }^{\delta }f\left(z\right)$ to the particular operators $L_p^{\mu }$ and $M_p^{\mu }$, as noted in Definition 3.

(ii) The limitation $\alpha \to 0$ returns the classical class of meromorphically starlike functions of order 0.

By selecting specific values for the parameters, we can recover the following special cases:

(i) ${\Sigma }_p\left(0,\mu ,\alpha \right)=\left\{f\in {\Sigma }_p:\left|{\displaystyle \frac{z{\left(L_p^{\mu }f\left(z\right)\right)}^{\prime }}{L_p^{\mu }f\left(z\right)}}+p\right|<\left|{\displaystyle \frac{z{\left(L_p^{\mu }f\left(z\right)\right)}^{\prime }}{L_p^{\mu }f\left(z\right)}}+2\alpha -p\right|\right\};$

(ii) ${\Sigma }_p\left(1,\mu ,\alpha \right)=\left\{f\in {\Sigma }_p:\left|{\displaystyle \frac{z{\left(M_p^{\mu }f\left(z\right)\right)}^{\prime }}{M_p^{\mu }f\left(z\right)}}+p\right|<\left|{\displaystyle \frac{z{\left(M_p^{\mu }f\left(z\right)\right)}^{\prime }}{M_p^{\mu }f\left(z\right)}}+2\alpha -p\right|\right\};$

(iii) ${\Sigma }_p\left(\delta ,0,\alpha \right)=\left\{f\in {\Sigma }_p:\left|{\displaystyle \frac{z{\left(I_p^{\delta }f\left(z\right)\right)}^{\prime }}{I_p^{\delta }f\left(z\right)}}+p\right|<\left|{\displaystyle \frac{z{\left(I_p^{\delta }f\left(z\right)\right)}^{\prime }}{I_p^{\delta }f\left(z\right)}}+2\alpha -p\right|\right\};$

(iv) ${\Sigma }_p\left(\delta ,\mu ,0\right)=\left\{f\in {\Sigma }_p:\left|{\displaystyle \frac{z{\left(I_{p,\mu }^{\delta }f\left(z\right)\right)}^{\prime }}{I_{p,\mu }^{\delta }f\left(z\right)}}+p\right|<\left|{\displaystyle \frac{z{\left(I_{p,\mu }^{\delta }f\left(z\right)\right)}^{\prime }}{I_{p,\mu }^{\delta }f\left(z\right)}}-p\right|\right\}.$

Also, we define the $N_{\sigma }–$neighborhood for $f\in {\Sigma }_p$, building on the work of Goodman [21], Ruscheweyh [10], and Altıntaş et al. [11,22].

Definition 5.

For $f\in {\Sigma }_p$ provided by (1) and for $\sigma \ge 0$, we defined the σ-neighborhood as

$$N_{\sigma }\left(f,u\right)=\left\{u:u\left(z\right)\in {\Sigma }_p,u\left(z\right)={\displaystyle \frac{1}{z^p}}+\sum _{k=1}^{\infty }{b}_k{z}^k\mathit{and}\sum _{k=1}^{\infty }k\left|a_k-b_k\right|\le \sigma \right\},$$

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

$$N_{\sigma }\left(e,u\right)=\left\{u:u\left(z\right)\in {\Sigma }_p,u\left(z\right)={\displaystyle \frac{1}{z^p}}+\sum _{k=1}^{\infty }{b}_k{z}^k\mathit{and}\sum _{k=1}^{\infty }k\left|b_k\right|\le \sigma \right\}.$$

(6)

For the univalent case, when $p=1$, this definition reduces to the $N_{\delta }–$neighborhood for the class Σ, a concept established by Elkhatib et al. ([23], with $q\to 1^{-}$).

Table 1 summarizes the key symbols and parameters frequently utilized in this study.

Table 1. Frequently used symbols.

Symbol	Description	Symbol	Description
p	Valency	$\phi (k,\delta ,\mu )$	Operator kernel
$\delta$	Operator order	$\mu$	Operator parameter
$\alpha$	Order of starlikeness	$N_{\sigma }$	Neighborhood of a function
c	First coefficient parameter	r	Radial variable $\left|z\right|$
E	Electric field	$\xi$	Neighborhood parameter
$\varphi$	Electrostatic potential	$\mathrm{\Phi }$	Dominant function

3. Principal Findings

Here, we establish the principal analytical characteristics of the function class ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right)$, introduced in Definition 4. Our first and most important finding is that a function must satisfy a necessary and sufficient coefficient condition to be classified in this class. This fundamental theorem is a powerful tool from which we can derive several essential properties. Specifically, we will obtain sharp coefficient estimates, distortion theorems that provide bounds on the function and its derivative, and other geometric properties such as radii of starlikeness and convexity. Indeed, we will investigate the structural properties, such as closure in convex linear combinations, the identification of extreme points, and a neighborhood theorem.

The following theorem characterizes the class ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right)$ using a necessary and sufficient coefficient condition.

Theorem 1.

Let f be given by (1). Then $f\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right)$ if and only if

$$\sum _{k=1}^{\infty }\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)a_k\le p-\alpha .$$

(7)

Proof. 

Assume inequality (7) holds. We will demonstrate that $f\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right)$. From (5) we have

$$\left|{\displaystyle \frac{z{\left(I_{p,\mu }^{\delta }f\left(z\right)\right)}^{\prime }}{I_{p,\mu }^{\delta }f\left(z\right)}}+p\right|-\left|{\displaystyle \frac{z{\left(I_{p,\mu }^{\delta }f\left(z\right)\right)}^{\prime }}{I_{p,\mu }^{\delta }f\left(z\right)}}+2\alpha -p\right|=$$

$$\left|z{\left(I_{p,\mu }^{\delta }f\left(z\right)\right)}^{\prime }+p{I}_{p,\mu }^{\delta }f\left(z\right)\right|-\left|z{\left(I_{p,\mu }^{\delta }f\left(z\right)\right)}^{\prime }+\left(2\alpha -p\right)I_{p,\mu }^{\delta }f\left(z\right)\right|=$$

$$\left|\sum _{k=1}^{\infty }\left(k+p\right)\phi \left(k,\delta ,\mu \right)a_k{z}^k\right|-\left|{\displaystyle \frac{2\left(\alpha -p\right)}{z^p}}+\sum _{k=1}^{\infty }\left(k+2\alpha -p\right)\phi \left(k,\delta ,\mu \right)a_k{z}^k\right|$$

$$\le \sum _{k=1}^{\infty }\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)a_k+\left(\alpha -p\right)\le 0.$$

Therefore, the maximum modulus theorem implies $f\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right).$ Conversely, assuming that $f\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right),$ we find that

$$\left|{\displaystyle \frac{\frac{z{\left(I_{p,\mu }^{\delta }f\left(z\right)\right)}^{\prime }}{I_{p,\mu }^{\delta }f\left(z\right)}+p}{\frac{z{\left(I_{p,\mu }^{\delta }f\left(z\right)\right)}^{\prime }}{I_{p,\mu }^{\delta }f\left(z\right)}+2\alpha -p}}\right|=\left|{\displaystyle \frac{{\sum }_{k=1}^{\infty }\left(k+p\right)\phi \left(k,\delta ,\mu \right)a_k{z}^k}{\frac{2\left(\alpha -p\right)}{z^p}+{\sum }_{k=1}^{\infty }\left(k+2\alpha -p\right)\phi \left(k,\delta ,\mu \right)a_k{z}^k}}\right|<1,$$

since $Re\left\{z\right\}\le \left|z\right|$ for all $z,$ we get

$$Re\left\{{\displaystyle \frac{{\sum }_{k=1}^{\infty }\left(k+p\right)\phi \left(k,\delta ,\mu \right)a_k{z}^k}{\frac{2\left(\alpha -p\right)}{z^p}+{\sum }_{k=1}^{\infty }\left(k+2\alpha -p\right)\phi \left(k,\delta ,\mu \right)a_k{z}^k}}\right\}<1.$$

As the inequality (7) is derived from the condition ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right)$ and holds for every $z\in U^{*}$, the maximum modulus principle implies that it is true on the closed disk. To determine the coefficient bound, we assume z is a positive real number ($z=r$, $0<r<1$). On the real axis, the expressions are real. Taking the limit as $z\to 1^{-}$ and noting that the coefficients $a_k$ are non-negative (ensuring that the series converges to the precise coefficient total), the inequality reduces immediately to the necessary condition (7). □

Theorem 1 leads to the following sharp bound on the coefficients $a_k$.

Corollary 1.

If $f\left(z\right)$ provided by (1) is in ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right)$, then for each $k\ge 1$, we have the sharp coefficient bound

$$a_k\le {\displaystyle \frac{p-\alpha }{\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)}},$$

where equality is attained for the extremal functions

$$f\left(z\right)={\displaystyle \frac{1}{z^p}}+{\displaystyle \frac{p-\alpha }{\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)}}{z}^k.$$

In many applications, it is beneficial to analyze a subclass of functions where the first coefficient, $a_1$, is fixed. Motivated by this, we introduce the subclass ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$ of ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right)$ as the set of functions

$$f\left(z\right)={\displaystyle \frac{1}{z^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}z+\sum _{k=2}^{\infty }{a}_k{z}^k,$$

(8)

for a constant c $(0\le c<1).$ Note that the limit $c\to 1^{-}$ in subclass ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$ defines functions with decreasing contributions from higher-order coefficients ($a_k\ge 2$).

We now establish the coefficient condition for functions in ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$.

Theorem 2.

For $f\left(z\right)$ provided by (8), =$f\left(z\right)\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$ if

$$\sum _{k=2}^{\infty }\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)a_k\le \left(p-\alpha \right)\left(1-c\right).$$

(9)

Proof. 

By putting the fixed coefficient

$$a_1={\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}\left(0<c<1\right),$$

into the general condition given by (7), we obtain the inequality

$$c+\sum _{k=2}^{\infty }{\displaystyle \frac{\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)}{\left(p-\alpha \right)}}{a}_k\le 1.$$

A simple rearranging of this statement yields the required result (9). The equality occurs for

$$f\left(z\right)={\displaystyle \frac{1}{z^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}z+{\displaystyle \frac{\left(p-\alpha \right)\left(1-c\right)}{\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)}}{z}^k\left(k\ge 2\right).$$

(10)

 □

Theorem 2 gives the following sharp estimate for the coefficients $a_k$ for $k\ge 2$.

Corollary 2.

If $f\left(z\right)\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$ given by (1), then for each $k\ge 2$, we have the sharp coefficient bound

$$a_k\le {\displaystyle \frac{\left(p-\alpha \right)\left(1-c\right)}{\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)}}\left(k\ge 2\right).$$

The equality holds for the extremal function $f\left(z\right)$ provided by (10).

Building on previous findings, we now set bounds on the sums of the coefficients. These estimations are critical to demonstrating the distortion theorems that follow.

Theorem 3.

If $f\left(z\right)\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right),$ then we have

$$\sum _{k=2}^{\infty }{a}_k\le {\displaystyle \frac{\left(p-\alpha \right)\left(1-c\right)}{\left(\alpha +2\right)\phi \left(2,\delta ,\mu \right)}},$$

(11)

and

$$\sum _{k=2}^{\infty }k{a}_k\le {\displaystyle \frac{2\left(p-\alpha \right)\left(1-c\right)}{\left(\alpha +2\right)\phi \left(2,\delta ,\mu \right)}}.$$

(12)

The results are sharp, with equality for both bounds being attained for the extremal function

$$f\left(z\right)={\displaystyle \frac{1}{z^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}z+{\displaystyle \frac{\left(p-\alpha \right)\left(1-c\right)}{\left(\alpha +2\right)\phi \left(2,\delta ,\mu \right)}}{z}^2.$$

Proof. 

Let $f\left(z\right)\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right).$ Then, using the inequality (9), we have

$$\left(\alpha +2\right)\phi \left(2,\delta ,\mu \right)\sum _{k=2}^{\infty }{a}_k\le \left(p-\alpha \right)\left(1-c\right),$$

which yields the desired inequality (11). To establish the second inequality, we again use condition (9), and we have

$$\phi \left(2,\delta ,\mu \right)\sum _{k=2}^{\infty }k{a}_k\le \left(p-\alpha \right)\left(1-c\right)-\alpha \phi \left(2,\delta ,\mu \right)\sum _{k=2}^{\infty }{a}_k.$$

Applying the result from (11), we now have

$$\phi \left(2,\delta ,\mu \right)\sum _{k=2}^{\infty }k{a}_k\le \left(p-\alpha \right)\left(1-c\right)-{\displaystyle \frac{\alpha \left(p-\alpha \right)\left(1-c\right)}{\left(\alpha +2\right)}},$$

and a simple rearrangement produces the desired inequality (12). □

We now apply the coefficient bounds from Theorem 3 to compute distortion properties for functions in the class ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$. The following theorem gives sharp upper and lower bounds for the modulus of $f\left(z\right)$.

Theorem 4.

For $f\left(z\right)\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$ and $0<\left|z\right|=r<1,$ we obtain

$${\displaystyle \frac{1}{r^p}}-{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}r-{\displaystyle \frac{\left(p-\alpha \right)\left(1-c\right)}{\left(\alpha +2\right)\phi \left(2,\delta ,\mu \right)}}{r}^2\le \left|f\left(z\right)\right|$$

$$\le {\displaystyle \frac{1}{r^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}r+{\displaystyle \frac{\left(p-\alpha \right)\left(1-c\right)}{\left(\alpha +2\right)\phi \left(2,\delta ,\mu \right)}}{r}^2.$$

(13)

These constraints are sharp because equality is established for the function $f\left(z\right)$ defined as

$$f\left(z\right)={\displaystyle \frac{1}{z^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}z+{\displaystyle \frac{\left(p-\alpha \right)\left(1-c\right)}{\left(\alpha +2\right)\phi \left(2,\delta ,\mu \right)}}{z}^2,$$

(14)

where both the upper and lower bounds are realized for a correctly chosen real z.

Proof. 

For $f\left(z\right)\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right),$ we get

$$\left|f\left(z\right)\right|=\left|{\displaystyle \frac{1}{z^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}z+\sum _{k=2}^{\infty }{a}_k{z}^k\right|$$

$$\le {\displaystyle \frac{1}{{\left|z\right|}^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}\left|z\right|+{\left|z\right|}^2\sum _{k=2}^{\infty }{a}_k$$

$$\le {\displaystyle \frac{1}{r^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}r+r^2\sum _{k=2}^{\infty }{a}_k,$$

and

$$\left|f\left(z\right)\right|=\left|{\displaystyle \frac{1}{z^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}z+\sum _{k=2}^{\infty }{a}_k{z}^k\right|$$

$$\ge {\displaystyle \frac{1}{{\left|z\right|}^p}}-{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}\left|z\right|-{\left|z\right|}^2\sum _{k=2}^{\infty }{a}_k$$

$$\ge {\displaystyle \frac{1}{r^p}}-{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}r-r^2\sum _{k=2}^{\infty }{a}_k.$$

By inserting the bound for the sum of coefficients of inequality (11), we obtain the desired result (13). □

Following a similar approach, we use the coefficient bounds of Theorem 3 to obtain the distortion bounds for the derivative $f^{\prime }\left(z\right)$.

Theorem 5.

For $f\left(z\right)\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$ and $0<\left|z\right|=r<1,$ we obtain

$${\displaystyle \frac{p}{r^{p+1}}}-{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}-{\displaystyle \frac{2\left(p-\alpha \right)\left(1-c\right)}{\left(\alpha +2\right)\phi \left(2,\delta ,\mu \right)}}r\le \left|f^{\prime }\left(z\right)\right|$$

$$\le {\displaystyle \frac{p}{r^{p+1}}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}+{\displaystyle \frac{2\left(p-\alpha \right)\left(1-c\right)}{\left(\alpha +2\right)\phi \left(2,\delta ,\mu \right)}}r.$$

(15)

These constraints are sharp because equality is established for the function $f\left(z\right)$ provided by (14).

Proof. 

For $f\left(z\right)\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right),$ we obtain

$$\left|f^{\prime }\left(z\right)\right|=\left|{\displaystyle \frac{-p}{z^{p+1}}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}+\sum _{k=2}^{\infty }k{a}_k{z}^{k-1}\right|$$

$$\le {\displaystyle \frac{p}{{\left|z\right|}^{p+1}}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}+\left|z\right|\sum _{k=2}^{\infty }k{a}_k$$

$$\le {\displaystyle \frac{p}{r^{p+1}}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}+r\sum _{k=2}^{\infty }k{a}_k$$

and

$$\left|f^{\prime }\left(z\right)\right|=\left|{\displaystyle \frac{-p}{z^{p+1}}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}+\sum _{k=2}^{\infty }k{a}_k{z}^{k-1}\right|$$

$$\ge {\displaystyle \frac{p}{r^{p+1}}}-{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}-r\sum _{k=2}^{\infty }k{a}_k$$

$$\ge {\displaystyle \frac{p}{{\left|z\right|}^{p+1}}}-{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}-\left|z\right|\sum _{k=2}^{\infty }k{a}_k,$$

Now, by applying the bound for the sum from Theorem 3 given in (12), we obtain the upper bound in the theorem’s assertion (15). □

The distortion bounds in Theorems 4 and 5 follow the predicted classical behavior. The growth in a function in ${\Sigma }_p$ is governed by the dominating singular term $1/r^p$, where $r=\left|z\right|\to 0^{+}$ is the radial variable. In addition, when the generalized operator $I_{p,\mu }^{\delta }$ is reduced to a classical operator, the bounds in these theorems collapse precisely to the known sharp bounds for those specific classes, demonstrating the coherence of our generalized conclusions.

We next find the sharp radius of the disk in which every function in ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$ is starlike for a given order v.

Theorem 6.

For $f\left(z\right)\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right),$ $f\left(z\right)$ is starlike of order v $\left(0\le v<p\right)$ in $\left|z\right|<r_1,$ where $r_1$ is the largest value for which

$${\displaystyle \frac{\left(3p-v\right)\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}{r}^2+{\displaystyle \frac{\left(p-\alpha \right)\left(k_0+2p-v\right)\left(1-c\right)}{\left(k_0+\alpha \right)\phi \left(k_0,\delta ,\mu \right)}}{r}^{k+1}\le p-v.$$

for $k\ge 2.$ This result is sharp because equality is established for the function $f\left(z\right)$ provided by (10).

Proof. 

To establish this, it is necessary to demonstrate that

$$\left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}+p\right|\le p-v,\left(\left|z\right|<r_1\right).$$

We observe that

$$\left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}+p\right|\le {\displaystyle \frac{{\displaystyle \frac{\left(p+1\right)\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}\left|z\right|+{\sum }_{k=2}^{\infty }\left(k+p\right)a_k{\left|z\right|}^k}{{\displaystyle \frac{1}{{\left|z\right|}^p}}-{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}\left|z\right|-{\sum }_{k=2}^{\infty }{a}_k{\left|z\right|}^k}}\le p-v.$$

(16)

Therefore, inequality (16) is satisfied for $\left|z\right|<r_1$ if the following condition holds:

$${\displaystyle \frac{\left(3p-v\right)\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}{r}_1^2+\sum _{k=2}^{\infty }\left(k+2p-v\right)a_k{r}_1^{k+1}\le p-v.$$

In view of the coefficient condition (9), we can choose the coefficients to be of the form

$$a_k\le {\displaystyle \frac{\left(p-\alpha \right)\left(1-c\right){\lambda }_k}{\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)}}\left(k\ge 2\right),$$

where ${\lambda }_k\ge 0$ and ${\sum }_{k=2}^{\infty }{\lambda }_k\le 1.$ For a fixed $r\in (0,1)$, we define $k_0=k_0\left(r\right)\ge 2$ as the index of the sequence

$${\displaystyle \frac{\left(p-\alpha \right)\left(k_0+2p-v\right)\left(1-c\right)}{\left(k_0+\alpha \right)\phi \left(k_0,\delta ,\mu \right)}}{r}^{k_0+1}$$

attains its maximum value. The presence of such an index $k_0$ is ensured since the coefficients are positive, and the behavior of the sequence for large k ensures that a maximum is reached within a bounded range of indices for any fixed $r\in (0,1)$. Therefore, we obtain

$$\sum _{k=2}^{\infty }\left(k+2p-v\right)a_k{r}^{k+1}\le {\displaystyle \frac{\left(p-\alpha \right)\left(k_0+2p-v\right)\left(1-c\right)}{\left(k_0+\alpha \right)\phi \left(k_0,\delta ,\mu \right)}}{r}^{k_0+1},$$

then f is starlike of order v in $\left|z\right|<r_1$ if the following condition is satisfied:

$${\displaystyle \frac{\left(3p-v\right)\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}{r}_1^2+{\displaystyle \frac{\left(p-\alpha \right)\left(k_0+2p-v\right)\left(1-c\right)}{\left(k_0+\alpha \right)\phi \left(k_0,\delta ,\mu \right)}}{r}_1^{k_0+1}\le p-v.$$

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

$${\displaystyle \frac{\left(3p-v\right)\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}{r}_0^2+{\displaystyle \frac{\left(p-\alpha \right)\left(k_0+2p-v\right)\left(1-c\right)}{\left(k_0+\alpha \right)\phi \left(k_0,\delta ,\mu \right)}}{r}_0^{k_0+1}=p-v.$$

This completes the proof, and the value $r_0$ represents the sharp radius of starlikeness of order v for the class $f\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right).$ □

Following the same line of reasoning used to determine the radius of starlikeness, we next determine the sharp radius of convexity for ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$.

Theorem 7.

For f $\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right),$ f is convex of order v $\left(0\le v<p\right)$ in $\left|z\right|<r_2,$ and $r_2$ is the largest value such that

$${\displaystyle \frac{\left(3p-v\right)\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}{r}^2+{\displaystyle \frac{k_0\left(p-\alpha \right)\left(k_0+2p-v\right)\left(1-c\right)}{\left(k_0+\alpha \right)\phi \left(k_0,\delta ,\mu \right)}}{r}^{k+1}\le p-v.$$

for $k\ge 2$. This result is sharp because equality is established for the function $f\left(z\right)$ provided by (10).

Proof. 

Using the proving technique of Theorem 6, we demonstrate that

$$\left|{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}+\left(p+1\right)\right|\le p-v,$$

and the result follows for $\left|z\right|<r_2$ by applying the coefficient bounds from Theorem 2. □

Now, we explore the closure properties of the class ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$.

Theorem 8.

The class ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$ is closed under a convex linear combination.

Proof. 

Let $f\left(z\right)$ and $h\left(z\right)$ be in ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$, $f\left(z\right)$ provided by (8), and let $h\left(z\right)$ be given by:

$$h\left(z\right)={\displaystyle \frac{1}{z^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}z+\sum _{k=2}^{\infty }{b}_k{z}^k\left(b_k\ge 0\right).$$

We need to show that the convex combination

$$G\left(z\right)=\zeta f\left(z\right)+\left(1-\zeta \right)h\left(z\right),$$

is also in the class for $0\le \zeta \le 1$. Writing the function $G\left(z\right)$ in its series form, we get

$$G\left(z\right)={\displaystyle \frac{1}{z^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}z+\sum _{k=2}^{\infty }\left\{\zeta a_k+\left(1-\zeta \right)b_k\right\}{z}^k.$$

A direct result of Theorem 2 is that

$$\sum _{k=2}^{\infty }\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)\left\{\zeta a_k+\left(1-\zeta \right)b_k\right\}\le \left(p-\alpha \right)\left(1-c\right).$$

Hence $G\left(z\right)\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right).$ □

We now find the extreme points of the class ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$. These points are essential because they create the entire class using convex combinations.

Theorem 9.

The extreme points of ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$ are $f_1\left(z\right)$ and $f_k\left(z\right)$, defined as follows:

$$f_1\left(z\right)={\displaystyle \frac{1}{z^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right),}}z,$$

(17)

and for $k\ge 1$,

$$f_k\left(z\right)={\displaystyle \frac{1}{z^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}z+\sum _{k=2}^{\infty }{\displaystyle \frac{\left(p-\alpha \right)\left(1-c\right)}{\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)}}{z}^k.$$

(18)

Consequently, a function $f\left(z\right)\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$ is equivalent to its expressibility as a convex linear combination of these extreme points:

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

(19)

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

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

(20)

Proof. 

First, suppose that $f\left(z\right)$ may be written as a convex linear combination of the kind provided in (19). By incorporating the definitions of the extreme points $f_k\left(z\right)$ from (17) and (18) into that sum, we get

$$f\left(z\right)={\displaystyle \frac{1}{z^p}}+{\displaystyle \frac{\left(p-\alpha \right)c}{\left(\alpha +1\right)\phi \left(1,\delta ,\mu \right)}}z+\sum _{k=2}^{\infty }{\displaystyle \frac{\left(p-\alpha \right)\left(1-c\right){\eta }_k}{\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)}}{z}^k.$$

Since

$$\sum _{k=2}^{\infty }{\displaystyle \frac{\left(p-\alpha \right)\left(1-c\right){\eta }_k}{\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)}}·{\displaystyle \frac{\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)}{\left(p-\alpha \right)\left(1-c\right)}}=\sum _{k=2}^{\infty }{\eta }_k=1-{\eta }_1\le 1,$$

according to Theorem 2, $f\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right).$ Conversely, consider $f\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$ and satisfies (12) for $k\ge 2$; then,

$${\eta }_k={\displaystyle \frac{\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)}{\left(p-\alpha \right)\left(1-c\right)}}{a}_k\le 1$$

and

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

With this, the proof of Theorem 9 is completed. □

Our final result for this section is a neighborhood theorem for the class ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$.

Theorem 10.

If $f\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right),$ then

$${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)\subset N_{\xi }\left(f,u\right),$$

(21)

where the parameter ξ is given by

$$\xi ={\displaystyle \frac{2\left(p-\alpha \right)\left(1-c\right)}{\left(\alpha +2\right)\phi \left(2,\delta ,\mu \right)}}.$$

(22)

The constant ξ is obtained using the sharp bound in Theorem 3, Equation (12).

Proof. 

For $f\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$, by simply applying the inequality (12) from Theorem 3 and in view of (8), we obtain (22). □

4. Applications to Fractional Electrodynamics

In this section, we demonstrate the practical utility of our theoretical framework developed in Section 2 and Section 3. Traditional fractional operators, such as Riemann–Liouville and Caputo operators, require a locally integrable initial condition $\varphi (x,0)$ near the singularity. However, electrostatic potentials with higher-order poles, such as $\varphi (x,0)\sim x^{-2}$, violate this condition, rendering these operators inapplicable in their standard form [24].

Higher-order singularities ($p>1$) provide a substantial issue in mathematical physics. Electrostatic systems, such as a charged hyperbolic protrusion or a dipole line (double layer), produce precisely integer-order poles of the form $\varphi \sim r^{-2}$ [25]. These are not approximations but actual solutions to Laplace’s equation ${\nabla }^2\varphi =0$ in non-smooth domains under Dirichlet or Neumann boundary conditions [25]. The Laurent expansion for such solutions is $\varphi \left(r\right)=r^{-p}q\left(r\right)$, with analytic and positive $q\left(r\right)$ near $r=0$ and non-negative Laurent coefficients. This structure categorizes them precisely as ${\Sigma }_p^{+}$. This alignment enables the straightforward application of the sharp coefficient bounds derived in Theorems 4 and 5.

Our operator $I_{p,\mu }^{\delta }$ is specifically designed to handle such singular solutions. We define a dominant function $\mathrm{\Phi }(x,t)$ whose coefficients are the absolute values of the series coefficients of the solution $\varphi (x,t)$. The Caputo fractional diffusion operator $D_t^{\delta }$ preserves the non-negativity of Laurent coefficients under initial conditions with non-negative $a_k\left(0\right)$ [26]. Additionally, our initial condition satisfies the sharp coefficient condition of Theorem 2:

$$\sum _{k=2}^{\infty }\left(k+\alpha \right)\phi \left(k,\delta ,\mu \right)a_k\left(t\right)\le \left(p-\alpha \right)\left(1-c\right),$$

for all $t\ge 0$. Thus, $\mathrm{\Phi }(x,t)\in {\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$. This allows us to apply the sharp distortion theorems (Theorems 4 and 5) to deduce physically meaningful and rigorous upper bounds for the solution and its derivative. This method translates abstract coefficient bounds into tangible engineering margins, resulting in a reliable and cautious approximation of field behavior near singularities.

To illustrate this methodology, we consider a canonical problem: the evolution of a singular potential governed by the one-dimensional fractional diffusion equation, a model of subdiffusive relaxation in dispersive media [27]. The specific problem is the fractional partial differential equation (FPDE):

$$D_t^{\delta }\varphi (x,t)=\kappa {\displaystyle \frac{{\partial }^2\varphi (x,t)}{\partial x^2}},(x>0,t>0),(0<\delta <1),$$

(23)

where $\kappa$ is the generalized diffusion coefficient, and $D_t^{\delta }$ is the fractional derivative of Caputo of order $\delta$ $\left(0<\delta <1\right)$, provided by [26]:

$$D_t^{\delta }u={\displaystyle \frac{1}{\mathrm{\Gamma }\left(1-\delta \right)}}{\int }_0^t{\left(t-\tau \right)}^{-\delta }{\displaystyle \frac{\partial u}{\partial \tau }}d\tau \left(0<\delta <1\right).$$

To ensure dimensional consistency in Equation (23), the units of the generalized diffusion coefficient $\kappa$ must be expressed in square millimeters per second to the power of delta $m{m}^2/s^{\delta }$, where $mm$ represents length L, and s represents time T.

The equation is subject to the following initial condition:

$$\varphi (x,0)=x^{-p}q\left(x\right),$$

(24)

where $p\ge 1$ is the order of the singularity, and $q\left(x\right)$ is an analytic and positive function near $x=0$. We interpret this as a radial problem on the punctured disk. Using $r=x/L$ (with $L=1$ mm), we map the physical domain $x\in (0,L)$ to $r\in (0,1)$. This scaling is crucial because it integrates the physical variable into the complex domain $U^{*}$, where function theory is defined, resulting in $\varphi (r,0)=r^{-p}q\left(r\right)\in {\Sigma }_p^{+}$. This mapping preserves the singularity structure and enables the straightforward application of our theoretical results to obtain theoretically derived bounds on the system’s behavior.

4.1. The Physical Model and Problem Formulation

We now identify the parameters for our model, which is designed to represent the subdiffusive relaxation of an electrostatic-like potential near a sharp, charged protrusion of a shape known to produce a second-order singularity in the potential field. Such configurations occur physically in systems such as the field as a result of a line of dipoles (a double layer) or a hyperbolic charge distribution [25].

The surrounding medium is treated as a polymer dielectric with unusual subdiffusive characteristics. We choose a fractional order of $\delta =0.7$, which aligns with relaxation dynamics in amorphous polymers [28], and a generalized diffusion coefficient of $\kappa =0.01$ mm/s, which represents diffusivity in such media. For the operator $I_{p,\mu }^{\delta }$, we work within the subclass ${\Sigma }_2^{+}(0.7,0,0,0)$, defined by the parameters

• $p=2$: corresponding to the dominant $r^{-2}$ singularity;
• $\delta =0.7$: matching the subdiffusive exponent of the medium;
• $\mu =0$: maximizing the sensitivity of the operator to the singularity;
• $\alpha =0$: assuring the least restrictive geometric condition for the extremal case;
• $c=0$: since the initial field has no linear component ($a_1=0$).

For the initial condition, we define the analytic function $q\left(x\right)$ as

$$q\left(x\right)=1+\epsilon x^4,\mathrm{where}\epsilon ={\displaystyle \frac{1}{\phi (2,0.7,0)}}.$$

(25)

Consequently, from (24), the specific initial condition is

$$\varphi (x,0)={\displaystyle \frac{1}{x^2}}+{\displaystyle \frac{1}{\phi (2,0.7,0)}}{x}^2.$$

(26)

This function belongs to subclass ${\Sigma }_2^{+}(0.7,0,0,0)$ and satisfies the coefficient condition of Theorem 2 with equality

$$2·\phi (2,0.7,0)·{\displaystyle \frac{1}{\phi (2,0.7,0)}}=2.$$

Combining these elements with Equation (23), the entire initial boundary value problem to be addressed is as follows:

$$D_t^{0.7}\varphi (x,t)=0.01{\displaystyle \frac{{\partial }^2\varphi (x,t)}{\partial x^2}},\varphi (x,0)={\displaystyle \frac{1}{x^2}}+{\displaystyle \frac{1}{\phi (2,0.7,0)}}{x}^2.$$

(27)

Our goal is twofold: first, to derive exact upper bounds for the magnitude of the potential $\left|\varphi \right(x,t\left)\right|$ and the electric field $|E(x,t)|=|-{\displaystyle \frac{\partial \varphi (x,t)}{\partial x}}|$; second, to demonstrate how these abstract coefficient-based bounds translate into conservative, life-saving engineering margins for high-voltage insulation systems operating near geometric singularities.

4.2. Solution Bounds via a Dominant Function

We now show our solution methods for the initial boundary value problem (27). Our approach focuses on constructing a dominating function $\mathrm{\Phi }(r,t)$ that majorizes $\varphi (x,t)$ rather than finding a closed-form solution. Using the operator $I_{2,0}^{0.7}$ and ensuring $\mathrm{\Phi }(r,t)$ remains within the subclass ${\Sigma }_2^{+}(0.7,0,0,0)$ for all $t\ge 0$, we can directly apply the sharp distortion theorems (Theorems 4 and 5) to obtain rigorous, time-invariant upper bounds for both the potential $\left|\varphi \right(x,t\left)\right|$ and the electric field $\left|E\right(x,t\left)\right|$.

4.2.1. Series Representation and Dominant Function Construction

Let us suppose that the solution to (27) enables a series representation of the form

$$\varphi (x,t)={\displaystyle \frac{1}{x^2}}+\sum _{k=1}^{\infty }{a}_k\left(t\right)x^k,$$

where the coefficients $a_k\left(t\right)$ are time-dependent. Using the initial condition (26), we get

$$a_2\left(0\right)={\displaystyle \frac{1}{\phi (2,0.7,0)}}\approx 7.83\times {10}^{-4},\mathrm{and}{a}_k\left(0\right)=0\mathrm{for}k\ne 2.$$

We now construct the dominant function $\mathrm{\Phi }(x,t)$ by taking the absolute values of the coefficients of $\varphi (x,t)$:

$$\mathrm{\Phi }(x,t)={\displaystyle \frac{1}{x^2}}+\sum _{k=1}^{\infty }\left|a_k\left(t\right)\right|x^k.$$

It follows immediately from the fundamental triangle inequality that

$$\left|\varphi \right(x,t\left)\right|\le \mathrm{\Phi }\left(\right|x|,t)\mathrm{for}\mathrm{all}x>0,t\ge 0.$$

Specifically, for $t=0$, $\mathrm{\Phi }(x,0)=\varphi (x,0)\in {\Sigma }_2^{+}(0.7,0,0,0)$. For $t>0$, we proceed with the premise that the dissipative nature of the Caputo-time fractional diffusion operator retains the non-negativity of the Laurent coefficients $a_k\left(t\right)$, which are initiated by the non-negative initial condition $a_k\left(0\right)$. This assures that the dominating function $\mathrm{\Phi }(x,t)$ remains within a subclass of ${\Sigma }_p^{+}$ for any $t\ge 0$, which is necessary for applying the distortion theorems.

4.2.2. Applications of Distortion Theorems

We now apply the distortion theorems to the dominant function $\mathrm{\Phi }(r,t)$, where $r=\left|x\right|$.

• Bound on Potential $\left|\varphi \right(x,t\left)\right|$

Applying Theorem 4, we have the following:

$$\left|\varphi (x,t)\right|\le \mathrm{\Phi }(r,t)\le {\displaystyle \frac{1}{r^2}}+{\displaystyle \frac{(2-0)(1-0)}{(0+2)\phi (2,0.7,0)}}{r}^2={\displaystyle \frac{1}{r^2}}+{\displaystyle \frac{1}{\phi (2,0.7,0)}}{r}^2.$$

(28)

• Bound on Electric Field $\left|E\right(x,t\left)\right|$

Applying Theorem 5, we have the following:

$$\left|E(x,t)\right|=\left|-{\displaystyle \frac{\partial \varphi }{\partial x}}\right|\le {\displaystyle \frac{2}{r^3}}+{\displaystyle \frac{2(2-0)(1-0)}{(0+2)\phi (2,0.7,0)}}r={\displaystyle \frac{2}{r^3}}+{\displaystyle \frac{2}{\phi (2,0.7,0)}}r.$$

(29)

These bounds apply for all $0<r<1$ and $t\ge 0$.

4.2.3. Numerical Evaluation and Physical Interpretation

From Equation (4), the constant $\phi (2,0.7,0)$ is given by

$$\phi (2,0.7,0)={\displaystyle \frac{{(2-0)}^{2+2}\mathrm{\Gamma }(2+2+0.7+1)}{\mathrm{\Gamma }(0.7+1)}}=1277.12.$$

From (28) and (29), for $r=0.1$ mm away from the wedge singularity, we obtain explicit, numerically tight constraints:

• Potential Bound

$$\left|\varphi (x,t)\right|\le 100+7.83\times {10}^{-6}\approx 100\mathrm{V}.$$

• Field Strength Bound

$$\left|E(x,t)\right|\le 2000+1.566\times {10}^{-4}\approx 2000\mathrm{V}/\mathrm{mm}.$$

These bounds ensure that the electric field magnitude does not exceed the projected limits, providing a conservative safety margin for preventing dielectric breakdown in high-voltage equipment near geometric singularities. This provides a direct, non-approximate translation of abstract operator theory into quantifiable, mathematically guaranteed engineering safety criteria.

5. Conclusions

This work successfully generalized the notion of a Rafid operator to meromorphic p-valent functions and introduced a new integral operator $I_{p,\mu }^{\delta }$. Using this operator, we defined and analyzed two novel subclasses ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha \right)$ and ${\Sigma }_p^{+}\left(\delta ,\mu ,\alpha ,c\right)$.

Our main contributions are comprehensive and sharp. We established a fundamental, necessary, and sufficient condition for membership in these classes, which provided the basis for drawing several accurate conclusions. These include sharp coefficient bounds, distortion theorems, radii of starlikeness and convexity of a given order, closure properties under convex linear combinations, extreme points, and neighborhood results. All findings are sharp and extremal functions are clearly characterized.

This paper presents a cohesive framework for extending and generalizing various known results in the geometric theory of meromorphic functions. This framework’s power is demonstrated by its successful application in Section 4, where it gives mathematically assured constraints for singular potentials in fractional electrodynamics, converting abstract operator theory to concrete safety criteria. The operator-based technique shown here is an effective tool for evaluating the geometric characteristics of meromorphic multivalent functions.

For future research, we will explore the potential application of this theoretical framework to address problems in complex differential equations or potential theory.