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14 pages, 569 KiB

Open AccessArticle

A New Subclass of Bi-Univalent Functions Defined by Subordination to Laguerre Polynomials and the (p,q)-Derivative Operator

by Mohammad El-Ityan, Tariq Al-Hawary, Basem Aref Frasin and Ibtisam Aldawish

Symmetry 2025, 17(7), 982; https://doi.org/10.3390/sym17070982 - 21 Jun 2025

Viewed by 430

Abstract

In this work, we introduce a new subclass of bi-univalent functions using the

(p, q)

-derivative operator and the concept of subordination to generalized Laguerre polynomials

L_{t}^{ς} (k)

, which satisfy the differential equation [...] Read more.

In this work, we introduce a new subclass of bi-univalent functions using the

(p, q)

-derivative operator and the concept of subordination to generalized Laguerre polynomials

L_{t}^{ς} (k)

, which satisfy the differential equation

k y^{″} + (1 + ς - k) y^{'} + t y = 0,

with

1 + ς > 0

k \in R

, and

t \geq 0

. We focus on functions that blend the geometric features of starlike and convex mappings in a symmetric setting. The main goal is to estimate the initial coefficients of functions in this new class. Specifically, we obtain sharp upper bounds for

| a_{2} |

and

| a_{3} |

and for the Fekete–Szegö functional

| a_{3} - η a_{2}^{2} |

for some real number

η

. In the final section, we explore several special cases that arise from our general results. These results contribute to the ongoing development of bi-univalent function theory in the context of

(p, q)

-calculus. Full article

(This article belongs to the Special Issue Applications of Special Functions in Complex Analysis and Their Symmetries)

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19 pages, 371 KiB

Open AccessArticle

New Results of Differential Subordination for a Specific Subclass of p-Valent Meromorphic Functions Involving a New Operator

by Nihad Hameed Shehab, Abdul Rahman S. Juma, Luminița-Ioana Cotîrlă and Daniel Breaz

Axioms 2024, 13(12), 878; https://doi.org/10.3390/axioms13120878 - 18 Dec 2024

Viewed by 658

Abstract

The present article aims to significantly improve geometric function theory by making an important contribution to p-valent meromorphic and analytic functions. It focuses on subordination, which describes the relationships of analytic functions. In order to achieve this, we utilize a technique that is based on the properties of differential subordination. This approach, which is one of the most recent developments in this field, may obtain a number of conclusions about differential subordination for p-valent meromorphic functions described by the new operator

{I H}_{p, q, s}^{j, p} (ν_{1}, n, α, l) J (ζ)

within the porous unit disk

Δ

. Numerous mathematical and practical issues involving orthogonal polynomials, such as system identification, signal processing, fluid dynamics, antenna technology, and approximation theory, can benefit from the results presented in this article. The knowledge and comprehension of the unit’s analytical functions and its interacting higher relations are also greatly expanded by this text. Full article

(This article belongs to the Special Issue Advances in Geometric Function Theory and Related Topics)

21 pages, 1949 KiB

Open AccessArticle

Computing Topological Descriptors of Prime Ideal Sum Graphs of Commutative Rings

by Esra Öztürk Sözen, Turki Alsuraiheed, Cihat Abdioğlu and Shakir Ali

Symmetry 2023, 15(12), 2133; https://doi.org/10.3390/sym15122133 - 30 Nov 2023

Cited by 10 | Viewed by 3097

Abstract

Let

n \geq 1

be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of

Z_{n}

, where [...] Read more.

Let

n \geq 1

be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of

Z_{n}

, where

n = p^{α}, p^{2} q, p^{2} q^{2}, p q r, p^{3} q,

p^{2} q r

, and

p q r s

for the different prime integers

p, q, r,

and s. Moreover, we construct M-polynomials and

C o M

-polynomials using the

P I S

-graph structure of

Z_{n}

to avoid the difficulty of computing the descriptors via formulas directly. Furthermore, we present a geometric comparison for representations of each surface obtained by M-polynomials and

C o M

-polynomials. Finally, we discuss the applicability of algebraic graphs to chemical graph theory. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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18 pages, 1114 KiB

Open AccessArticle

Diverse Properties and Approximate Roots for a Novel Kinds of the (p,q)-Cosine and (p,q)-Sine Geometric Polynomials

by Sunil Kumar Sharma, Waseem Ahmad Khan, Cheon-Seoung Ryoo and Ugur Duran

Mathematics 2022, 10(15), 2709; https://doi.org/10.3390/math10152709 - 31 Jul 2022

Cited by 3 | Viewed by 1639

Abstract

Utilizing

(p, q)

-numbers and

(p, q)

-concepts, in 2016, Duran et al. considered

(p, q)

-Genocchi numbers and polynomials,

(p, q)

-Bernoulli numbers and polynomials and

(p, q)

-Euler polynomials and numbers and provided multifarious formulas and [...] Read more.

Utilizing

(p, q)

-numbers and

(p, q)

-concepts, in 2016, Duran et al. considered

(p, q)

-Genocchi numbers and polynomials,

(p, q)

-Bernoulli numbers and polynomials and

(p, q)

-Euler polynomials and numbers and provided multifarious formulas and properties for these polynomials. Inspired and motivated by this consideration, many authors have introduced

(p, q)

-special polynomials and numbers and have described some of their properties and applications. In this paper, using the

(p, q)

-cosine polynomials and

(p, q)

-sine polynomials, we consider a novel kinds of

(p, q)

-extensions of geometric polynomials and acquire several properties and identities by making use of some series manipulation methods. Furthermore, we compute the

(p, q)

-integral representations and

(p, q)

-derivative operator rules for the new polynomials. Additionally, we determine the movements of the approximate zerosof the two mentioned polynomials in a complex plane, utilizing the Newton method, and we illustrate them using figures. Full article

(This article belongs to the Special Issue Q-differential/Difference Equations and Related Applications)

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