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19 pages, 532 KB

Open AccessArticle

Third-Order Fuzzy Differential Results for Meromorphic Functions Using a Linear Operator: Subordination and Superordination

by Mays S. Abdul Ameer, Abdul Rahman S. Juma and Hassan Hussien Ebrahim

Symmetry 2026, 18(3), 413; https://doi.org/10.3390/sym18030413 - 27 Feb 2026

Viewed by 131

Abstract

While over a hundred articles discuss second-order differential inequalities and subordinations in the complex plane, very few address the relatively unexplored classes of third-order fuzzy differential subordination and superordination. This paper builds upon the recently proposed concepts of third-order fuzzy differential subordination and [...] Read more.

While over a hundred articles discuss second-order differential inequalities and subordinations in the complex plane, very few address the relatively unexplored classes of third-order fuzzy differential subordination and superordination. This paper builds upon the recently proposed concepts of third-order fuzzy differential subordination and superordination, which are developed using a linear operator and a meromorphic function. By applying techniques based on the fundamental notion of admissible functions, we begin by defining the appropriate class of such functions necessary for deriving new results in third-order fuzzy differential subordination. The study reveals the establishment of sandwich-type theorems, linking these new findings with established methods in third-order fuzzy differentiation and superordination theory. Full article

(This article belongs to the Section Mathematics)

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16 pages, 291 KB

Open AccessArticle

Normal Criterion for Families of Meromorphic Functions and Shared Functions

by Ai Huang and Jinhua Yang

Mathematics 2026, 14(2), 353; https://doi.org/10.3390/math14020353 - 20 Jan 2026

Viewed by 239

Abstract

This paper broadens the scope of existing research: the shared value is generalized from a non-zero finite complex number to a non-identically zero holomorphic function, the order of the derivative is extended from the first order to an arbitrary k-th order, and [...] Read more.

This paper broadens the scope of existing research: the shared value is generalized from a non-zero finite complex number to a non-identically zero holomorphic function, the order of the derivative is extended from the first order to an arbitrary k-th order, and the constraint condition on the polynomial H is simplified to

deg H \geq 2

. A more general normality criterion for families of meromorphic functions involving the sharing of differential polynomials is proved. Let D be a domain,

F

be a family of meromorphic functions in D, and

P (z)

be a non-identically zero holomorphic function in D. If for any

f, g \in F

, the differential polynomials

H (f) f^{(k)}

and

H (g) g^{(k)}

share

P (z)

in D, then

F

is normal in D. Full article

(This article belongs to the Section C4: Complex Analysis)

19 pages, 327 KB

Open AccessFeature PaperArticle

Analytic Continuation of the Hurwitz Transform

by Namhoon Kim

Mathematics 2026, 14(2), 271; https://doi.org/10.3390/math14020271 - 10 Jan 2026

Viewed by 373

Abstract

An integral over the unit interval of the product of a given function and the Hurwitz zeta function is known as the Hurwitz transform of the function. We give sufficient conditions on the function for the Hurwitz transform to continue meromorphically to a [...] Read more.

An integral over the unit interval of the product of a given function and the Hurwitz zeta function is known as the Hurwitz transform of the function. We give sufficient conditions on the function for the Hurwitz transform to continue meromorphically to a larger region and to the complex plane. Integral representations for the meromorphic extension of the Hurwitz transform are given, and some new integral identities involving the Hurwitz zeta function are derived. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

17 pages, 307 KB

Open AccessArticle

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

by Aya F. Elkhatib, Atef F. Hashem, Adela O. Mostafa and Mohammed M. Tharwat

Symmetry 2025, 17(11), 1837; https://doi.org/10.3390/sym17111837 - 2 Nov 2025

Viewed by 361

Abstract

This study introduces a new integral operator

I_{p, μ}^{δ}

that extends the traditional Rafid operator to meromorphic p-valent functions. Using this operator, we define and investigate two new subclasses:

Σ_{p}^{+} (δ, μ, α)

, consisting [...] Read more.

This study introduces a new integral operator

I_{p, μ}^{δ}

that extends the traditional Rafid operator to meromorphic p-valent functions. Using this operator, we define and investigate two new subclasses:

Σ_{p}^{+} (δ, μ, α)

, consisting of functions with nonnegative coefficients, and

Σ_{p}^{+} (δ, μ, α, c)

, 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

Σ_{p}^{+} (δ, μ, α, c)

, 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. Full article

(This article belongs to the Special Issue Symmetry in Complex Analysis Operators Theory)

14 pages, 305 KB

Open AccessArticle

Some Properties of Meromorphic Functions Defined by the Hurwitz–Lerch Zeta Function

by Ekram E. Ali, Rabha M. El-Ashwah, Nicoleta Breaz and Abeer M. Albalahi

Mathematics 2025, 13(21), 3430; https://doi.org/10.3390/math13213430 - 28 Oct 2025

Cited by 1 | Viewed by 544

Abstract

The findings of this study are connected with geometric function theory and were acquired using subordination-based techniques in conjunction with the Hurwitz–Lerch Zeta function. We used the Hurwitz–Lerch Zeta function to investigate certain properties of multivalent meromorphic functions. The primary objective of this [...] Read more.

The findings of this study are connected with geometric function theory and were acquired using subordination-based techniques in conjunction with the Hurwitz–Lerch Zeta function. We used the Hurwitz–Lerch Zeta function to investigate certain properties of multivalent meromorphic functions. The primary objective of this study is to provide an investigation on the argument properties of multivalent meromorphic functions in a punctured open unit disc and to obtain some results for its subclass. Full article

(This article belongs to the Special Issue Current Topics in Geometric Function Theory, 2nd Edition)

16 pages, 374 KB

Open AccessArticle

An Extended Complex Method to Solve the Predator–Prey Model

by Hongqiang Tu and Guoqiang Dang

Axioms 2025, 14(10), 758; https://doi.org/10.3390/axioms14100758 - 10 Oct 2025

Viewed by 706

Abstract

Through transformation and utilizing a novel extended complex method combining with the Weierstrass factorization theorem, Wiman–Valiron theory and the Painlevé test, new non-constant meromorphic solutions were constructed for the predator–prey model. These meromorphic solutions contain the rational solutions, exponential solutions, elliptic solutions, and [...] Read more.

Through transformation and utilizing a novel extended complex method combining with the Weierstrass factorization theorem, Wiman–Valiron theory and the Painlevé test, new non-constant meromorphic solutions were constructed for the predator–prey model. These meromorphic solutions contain the rational solutions, exponential solutions, elliptic solutions, and transcendental entire function solutions of infinite order in the complex plane. The exact solutions contribute to understanding the predator–prey model from the perspective of complex differential equations. In fact, the presented synthesis method provides a new technology for studying some systems of partial differential equations. Full article

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9 pages, 348 KB

Open AccessArticle

A Two-Stage Numerical Algorithm for the Simultaneous Extraction of All Zeros of Meromorphic Functions

by Ivan K. Ivanov and Stoil I. Ivanov

AppliedMath 2025, 5(4), 138; https://doi.org/10.3390/appliedmath5040138 - 6 Oct 2025

Viewed by 652

Abstract

In this paper, we present an effective two-stage numerical algorithm for the simultaneous finding of all roots of meromorphic functions in a region within the complex plane. At the first stage, we construct a polynomial with the same roots as the ones of [...] Read more.

In this paper, we present an effective two-stage numerical algorithm for the simultaneous finding of all roots of meromorphic functions in a region within the complex plane. At the first stage, we construct a polynomial with the same roots as the ones of the considered function; at the next step, we apply some method for the simultaneous approximation of its roots. To show the efficiency and applicability of our algorithm together with its advantages over the classical Newton, Halley and Chebyshev’s iterative methods, we conduct three numerical examples, where we apply it to two test functions and to an important engineering problem. Full article

(This article belongs to the Special Issue Advances in Iterative Methods and Stability Analysis for Solving Nonlinear Problems)

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19 pages, 2685 KB

Open AccessArticle

Sharp Bounds and Electromagnetic Field Applications for a Class of Meromorphic Functions Introduced by a New Operator

by Abdelrahman M. Yehia, Atef F. Hashem, Samar M. Madian and Mohammed M. Tharwat

Axioms 2025, 14(9), 684; https://doi.org/10.3390/axioms14090684 - 5 Sep 2025

Cited by 1 | Viewed by 617

Abstract

In this paper, we present a new integral operator that acts on a class of meromorphic functions on the punctured unit disc

U^{*}

. This operator enables the definition of a new subclass of meromorphic univalent functions. We obtain sharp bounds for [...] Read more.

In this paper, we present a new integral operator that acts on a class of meromorphic functions on the punctured unit disc

U^{*}

. This operator enables the definition of a new subclass of meromorphic univalent functions. We obtain sharp bounds for the Fekete–Szegö inequality and the second Hankel determinant for this class. The theoretical approach is based on differential subordination. Furthermore, we link these theoretical insights to applications in 2D electromagnetic field theory by outlining a physical framework in which the operator functions as a field transformation kernel. We show that the operator’s parameters correspond to physical analogs of field regularization and spectral redistribution, and we use subordination theory to simulate the design of vortex-free fields. The findings provide new insights into the interaction between geometric function theory and physical field modeling. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 4th Edition)

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18 pages, 3211 KB

Open AccessArticle

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

by Aya F. Elkhatib, Atef F. Hashem, Adela O. Mostafa and Mohammed M. Tharwat

Axioms 2025, 14(8), 620; https://doi.org/10.3390/axioms14080620 - 8 Aug 2025

Cited by 1 | Viewed by 677

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)

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15 pages, 307 KB

Open AccessArticle

Fuzzy Treatment for Meromorphic Classes of Admissible Functions Connected to Hurwitz–Lerch Zeta Function

by Ekram E. Ali, Rabha M. El-Ashwah, Abeer M. Albalahi and Rabab Sidaoui

Axioms 2025, 14(7), 523; https://doi.org/10.3390/axioms14070523 - 8 Jul 2025

Cited by 3 | Viewed by 562

Abstract

Fuzzy differential subordinations, a notion taken from fuzzy set theory and used in complex analysis, are the subject of this paper. In this work, we provide an operator and examine the characteristics of meromorphic functions in the punctured open unit disk that are [...] Read more.

Fuzzy differential subordinations, a notion taken from fuzzy set theory and used in complex analysis, are the subject of this paper. In this work, we provide an operator and examine the characteristics of meromorphic functions in the punctured open unit disk that are related to a class of complex parameter operators. Complex analysis ideas from geometric function theory are used to derive fuzzy differential subordination conclusions. Due to the compositional structure of the operator, some pertinent classes of admissible functions are studied through the application of fuzzy differential subordination. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)

18 pages, 546 KB

Open AccessArticle

Third-Order Differential Subordination Results for Meromorphic Functions Associated with the Inverse of the Legendre Chi Function via the Mittag-Leffler Identity

by Adel Salim Tayyah, Waggas Galib Atshan and Georgia Irina Oros

Mathematics 2025, 13(13), 2089; https://doi.org/10.3390/math13132089 - 25 Jun 2025

Cited by 7 | Viewed by 743

Abstract

In this paper, we derive novel results concerning third-order differential subordinations for meromorphic functions, utilizing a newly defined linear operator that involves the inverse of the Legendre chi function in conjunction with the Mittag-Leffler identity. To establish these results, we introduce several families [...] Read more.

In this paper, we derive novel results concerning third-order differential subordinations for meromorphic functions, utilizing a newly defined linear operator that involves the inverse of the Legendre chi function in conjunction with the Mittag-Leffler identity. To establish these results, we introduce several families of admissible functions tailored to this operator and formulate sufficient conditions under which the subordinations hold. Our study presents three fundamental theorems that extend and generalize known results in the literature. Each theorem is accompanied by rigorous proofs and further supported by corollaries and illustrative examples that validate the applicability and sharpness of the derived results. In particular, we highlight special cases and discuss their implications through both analytical evaluations and graphical interpretations, demonstrating the strength and flexibility of our framework. This work contributes meaningfully to the field of geometric function theory by offering new insights into the behavior of third-order differential operators acting on p-valent meromorphic functions. Furthermore, the involvement of the Mittag-Leffler function positions the results within the broader context of fractional calculus, suggesting potential for applications in the mathematical modeling of complex and nonlinear phenomena. We hope this study stimulates further research in related domains. Full article

(This article belongs to the Special Issue Current Topics in Geometric Function Theory, 2nd Edition)

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19 pages, 1200 KB

Open AccessFeature PaperArticle

A Subclass of Meromorphic Multivalent Functions Generated by a Symmetric q-Difference Operator

by Vasile-Aurel Caus

Mathematics 2025, 13(11), 1797; https://doi.org/10.3390/math13111797 - 28 May 2025

Cited by 1 | Viewed by 726

Abstract

This paper presents a novel symmetric q-analogue differential operator designed for meromorphic multivalent functions analytic in the punctured open unit disk. Employing this operator, a new family of meromorphic multivalent functions is proposed and examined in this work. A detailed investigation of [...] Read more.

This paper presents a novel symmetric q-analogue differential operator designed for meromorphic multivalent functions analytic in the punctured open unit disk. Employing this operator, a new family of meromorphic multivalent functions is proposed and examined in this work. A detailed investigation of this newly defined class of meromorphic multivalent functions is presented, highlighting key geometric characteristics, including sufficiency criteria, coefficient inequalities, distortion and growth behavior, as well as the radii of starlikeness and convexity. Full article

(This article belongs to the Section C4: Complex Analysis)

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23 pages, 340 KB

Open AccessArticle

Third-Order Fuzzy Subordination and Superordination on Analytic Functions on Punctured Unit Disk

by Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah and Abeer M. Albalahi

Axioms 2025, 14(5), 378; https://doi.org/10.3390/axioms14050378 - 17 May 2025

Cited by 1 | Viewed by 574

Abstract

This work’s theorems and corollaries present new third-order fuzzy differential subordination and superordination results developed by using a novel convolution linear operator involving the Gaussian hypergeometric function and a previously studied operator. The paper reveals methods for finding the best dominant and best [...] Read more.

This work’s theorems and corollaries present new third-order fuzzy differential subordination and superordination results developed by using a novel convolution linear operator involving the Gaussian hypergeometric function and a previously studied operator. The paper reveals methods for finding the best dominant and best subordinant for the third-order fuzzy differential subordinations and superordinations, respectively. The investigation concludes with the assertion of sandwich-type theorems connecting the conclusions of the studies conducted using the particular methods of the theories of the third-order fuzzy differential subordination and superordination, respectively. Full article

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

13 pages, 285 KB

Open AccessArticle

Majorization Problems for Subclasses of Meromorphic Functions Defined by the Generalized

q

-Sălăgean Operator

by Ekram E. Ali, Rabha M. El-Ashwah, Teodor Bulboacă and Abeer M. Albalahi

Mathematics 2025, 13(10), 1612; https://doi.org/10.3390/math13101612 - 14 May 2025

Viewed by 599

Abstract

Using the generalized

q

-Sălăgean operator, we introduce a new class of meromorphic functions in a punctured unit disk

U^{*}

and investigate a majorization problem associated with this class. The principal tool employed in this analysis is the recently established

q

-Schwarz–Pick [...] Read more.

Using the generalized

q

-Sălăgean operator, we introduce a new class of meromorphic functions in a punctured unit disk

U^{*}

and investigate a majorization problem associated with this class. The principal tool employed in this analysis is the recently established

q

-Schwarz–Pick lemma. We investigate a majorization problem for meromorphic functions when the functions of the right hand side of the majorization belongs to this class. The main tool for this investigation is the generalization of Nehari’s lemma for the Jackson’s

q

-difference operator

\partial_{q}

given recently by Adegani et al. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

14 pages, 323 KB

Open AccessArticle

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

by Mohammad El-Ityan, Luminita-Ioana Cotîrlă, Tariq Al-Hawary, Suha Hammad, Daniel Breaz and Rafid Buti

Symmetry 2025, 17(5), 728; https://doi.org/10.3390/sym17050728 - 9 May 2025

Cited by 4 | Viewed by 817

Abstract

In this paper, a novel class of meromorphic functions associated with the Mittag–Leffler function

E_{μ, ϑ} (z)

is introduced using the Hilbert space operator. In the punctured symmetric domain

⋓^{*}

, essential properties of this class are systematically [...] Read more.

In this paper, a novel class of meromorphic functions associated with the Mittag–Leffler function

E_{μ, ϑ} (z)

is introduced using the Hilbert space operator. In the punctured symmetric domain

⋓^{*}

, essential properties of this class are systematically investigated. These properties include coefficient inequalities, growth and distortion bounds, as well as weighted and arithmetic mean estimates. Furthermore, the extreme points and radii of geometric properties such as close-to-convexity, starlikeness, and convexity are analyzed in detail. Additionally, the Hadamard product (or convolution) is explored to demonstrate the algebraic structure and stability of the introduced function class under this operation. Integral mean inequalities are also established to provide further insights into the behavior of these functions within the given domain. Full article

(This article belongs to the Special Issue Symmetry and Its Applications in Complex Analysis by the Means of Special Functions)

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