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13 pages, 534 KB

Open AccessArticle

Certain Geometric Investigations of Three Normalized Bessel-Type Functions of a Complex Variable

by Rabab Alyusof, Shams Alyusof, Rabha M. El-Ashwah and Alaa H. El-Qadeem

Mathematics 2025, 13(23), 3888; https://doi.org/10.3390/math13233888 - 4 Dec 2025

We recall the normalized forms for the three Bessel-type functions; these functions are the Bessel function, Lommel function, and Struve function of the first kind. By using convolution, we define normalized forms. The essential purpose is to introduce necessary and sufficient bounds of [...] Read more.

We recall the normalized forms for the three Bessel-type functions; these functions are the Bessel function, Lommel function, and Struve function of the first kind. By using convolution, we define normalized forms. The essential purpose is to introduce necessary and sufficient bounds of these normalized functions so these functions are starlike and convex of order

γ

and type

δ

. Full article

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

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17 pages, 340 KB

Open AccessArticle

Certain Subclasses of Te-Univalent Functions Subordinate to q-Bernoulli Polynomials

by Sondekola Rudra Swamy, A. Alameer, Basem Aref Frasin and Savithri Shashidhar

Mathematics 2025, 13(23), 3841; https://doi.org/10.3390/math13233841 - 30 Nov 2025

Viewed by 75

Abstract

The present work centers on the significance of q-calculus in geometric function theory and its expanding applications within the domain of Te-univalent functions, especially those associated with special polynomials like the q-Bernoulli polynomials. Motivated by recent interest in these polynomials, our [...] Read more.

The present work centers on the significance of q-calculus in geometric function theory and its expanding applications within the domain of Te-univalent functions, especially those associated with special polynomials like the q-Bernoulli polynomials. Motivated by recent interest in these polynomials, our study introduces and analyzes a generalized subclass of Te-univalent functions that intimately relate to q-Bernoulli polynomials. For this new family, we establish explicit bounds for

| d_{2} |

and

| d_{3} |

, and provide estimates for the Fekete–Szegö functional

| d_{3} - ξ d_{2}^{2} |, ξ \in R

. Our findings contribute new results and demonstrate meaningful connections to prior work involving Te-univalent and subordinate functions, thereby broadening and integrating various strands of the existing literature. Full article

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

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

Viewed by 381

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)

20 pages, 840 KB

Open AccessArticle

Sharp Functional Inequalities for Starlike and Convex Functions Defined via a Single-Lobed Elliptic Domain

by Adel Salim Tayyah, Sarem H. Hadi, Abdullah Alatawi, Muhammad Abbas and Ovidiu Bagdasar

Mathematics 2025, 13(21), 3367; https://doi.org/10.3390/math13213367 - 22 Oct 2025

Cited by 1 | Viewed by 355

Abstract

In this paper, we introduce two novel subclasses of analytic functions, namely, starlike and convex functions of Ma–Minda-type, associated with a newly proposed domain. We set sharp bounds on the basic coefficients of these classes and provide sharp estimates of the second- and [...] Read more.

In this paper, we introduce two novel subclasses of analytic functions, namely, starlike and convex functions of Ma–Minda-type, associated with a newly proposed domain. We set sharp bounds on the basic coefficients of these classes and provide sharp estimates of the second- and third-order Hankel determinants, demonstrating the power of our analytic approach, the clarity of its results, and its applicability even in unconventional domains. Full article

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

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23 pages, 345 KB

Open AccessFeature PaperArticle

On Certain Subclasses of Analytic Functions Associated with a Symmetric q-Differential Operator

by Vasile-Aurel Caus

Mathematics 2025, 13(17), 2860; https://doi.org/10.3390/math13172860 - 4 Sep 2025

Viewed by 713

Abstract

This paper explores a class of analytic functions defined in the open unit disk by means of a symmetric q-differential operator. In the first part, we derive sufficient conditions for functions to belong to a subclass associated with this operator, using inequalities [...] Read more.

This paper explores a class of analytic functions defined in the open unit disk by means of a symmetric q-differential operator. In the first part, we derive sufficient conditions for functions to belong to a subclass associated with this operator, using inequalities involving their coefficients. Additionally, we establish several inclusion relations between these subclasses, obtained by varying the defining parameters. In the second part, we focus on differential subordination and superordination for functions transformed by the operator. We provide sufficient conditions under which such functions are subordinate or superordinate to univalent functions, and we determine the best dominant and best subordinant in specific cases. These results are complemented by several corollaries that highlight particular instances of the main theorems. Furthermore, we present a sandwich-type result that brings together the subordination and superordination frameworks in a unified analytic statement. Full article

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

18 pages, 608 KB

Open AccessArticle

The Geometric Characterizations of the Ramanujan-Type Entire Function

by Khaled Mehrez and Abdulaziz Alenazi

Mathematics 2025, 13(14), 2301; https://doi.org/10.3390/math13142301 - 18 Jul 2025

Viewed by 581

Abstract

In the present paper, we present certain geometric properties, such as starlikeness, convexity of order

η (0 \leq η < 1)

, and close-to-convexity, in an open unit disk of the normalized form of Ramanujan-type entire functions. As a consequence, a [...] Read more.

In the present paper, we present certain geometric properties, such as starlikeness, convexity of order

η (0 \leq η < 1)

, and close-to-convexity, in an open unit disk of the normalized form of Ramanujan-type entire functions. As a consequence, a specific range of parameters is derived such that this function belongs to Hardy spaces

H^{\infty}

and

H^{r} .

Finally, as an application, we present the monotonicity property of the Ramanujan-type entire function using the method of subordination factor sequences. Full article

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

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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 3 | Viewed by 505

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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9 pages, 265 KB

Open AccessArticle

Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions

by Tariq Al-Hawary, Basem Frasin and Ibtisam Aldawish

Mathematics 2025, 13(12), 2029; https://doi.org/10.3390/math13122029 - 19 Jun 2025

Viewed by 612

Abstract

In this paper, we demonstrate a relationship between a generalized distribution series and a comprehensive subclass of analytic functions. The primary aim of this study is to determine a necessary and sufficient condition for the generalized distribution series [...] Read more.

In this paper, we demonstrate a relationship between a generalized distribution series and a comprehensive subclass of analytic functions. The primary aim of this study is to determine a necessary and sufficient condition for the generalized distribution series

E_{ϕ}^{*} (ς, z)

to belong to the inclusive subclass

Π_{η} (Q_{3}, Q_{2}, Q_{1}, Q_{0})

. Necessary and sufficient conditions are also given for the generalized distribution series

E_{ϕ}^{*} (ς, z) ℏ

and the integral operator

J_{ς}^{ϕ} (z)

to be in the inclusive subclass

Π_{η} (Q_{3}, Q_{2}, Q_{1}, 0)

. Further, we provide a number of corollaries, which improve the existing ones that are available in some recent studies. The results presented here not only improve the earlier studies, but also give rise to a number of new results for particular choices of

Q_{3}, Q_{2}, Q_{1}

and

Q_{0}

. Full article

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

16 pages, 613 KB

Open AccessArticle

A Study of Certain Geometric Properties and Hardy Spaces of the Normalized Miller-Ross Function

by Muhammad Abubakr, Mohsan Raza, Abdulaziz Alenazi and Khaled Mehrez

Mathematics 2025, 13(12), 1919; https://doi.org/10.3390/math13121919 - 8 Jun 2025

Viewed by 569

Abstract

The main objective of this research is to investigate specific sufficiency criteria for the strongly starlikeness, strongly convexity, starlikeness, convexity and pre-starlikeness of the normalized Miller-Ross function. Furthermore, we establish sufficient conditions under which the normalized Miller-Ross function belongs to Hardy spaces and [...] Read more.

The main objective of this research is to investigate specific sufficiency criteria for the strongly starlikeness, strongly convexity, starlikeness, convexity and pre-starlikeness of the normalized Miller-Ross function. Furthermore, we establish sufficient conditions under which the normalized Miller-Ross function belongs to Hardy spaces and the class-bounded analytic functions. Some of the various results which are derived in this paper are presumably new and their significance is illustrated through several interesting examples. Full article

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

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19 pages, 322 KB

Open AccessArticle

Application on Fuzzy Third-Order Subordination and Superordination Connected with Lommel Function

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

Mathematics 2025, 13(12), 1917; https://doi.org/10.3390/math13121917 - 8 Jun 2025

Viewed by 502

Abstract

This work is based on the recently introduced concepts of third-order fuzzy differential subordination and its dual, third-order fuzzy differential superordination. In order to obtain the new results that add to the development of the newly initiated lines of research, a new operator [...] Read more.

This work is based on the recently introduced concepts of third-order fuzzy differential subordination and its dual, third-order fuzzy differential superordination. In order to obtain the new results that add to the development of the newly initiated lines of research, a new operator is defined here using the concept of convolution and the normalized Lommel function. The methods focusing on the basic concept of admissible function are employed. Hence, the investigation of new third-order fuzzy differential subordination results starts with the definition of the suitable class of admissible functions. The first theorems discuss third-order fuzzy differential subordinations involving the newly introduced operator. The following result shows the conditions needed such that the fuzzy best dominant can be found for a third-order fuzzy differential subordination. Next, dual results are obtained by employing the methods of third-order fuzzy differential superordination based on the same concept of an admissible function. A suitable class of admissible functions is introduced and new third-order fuzzy differential superordinations are obtained, showing how the best subordinant can be obtained under certain restrictions. As a conclusion of this study, sandwhich-type results are derived, linking the outcome of the two dual fuzzy theories. Full article

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

12 pages, 671 KB

Open AccessArticle

Inequalities of a Class of Analytic Functions Involving Multiplicative Derivative

by Kadhavoor R. Karthikeyan, Daniel Breaz, Gangadharan Murugusundaramoorthy and Ganapathi Thirupathi

Mathematics 2025, 13(10), 1606; https://doi.org/10.3390/math13101606 - 14 May 2025

Viewed by 579

Abstract

Using the concepts of multiplicative calculus and subordination of analytic functions, we define a new class of starlike bi-univalent functions based on a symmetric operator, which involved the three parameter Mittag-Leffler function. Estimates for the initial coefficients and Fekete–Szegő inequalities of the defined [...] Read more.

Using the concepts of multiplicative calculus and subordination of analytic functions, we define a new class of starlike bi-univalent functions based on a symmetric operator, which involved the three parameter Mittag-Leffler function. Estimates for the initial coefficients and Fekete–Szegő inequalities of the defined function classes are determined. Moreover, special cases of the classes have been discussed and stated as corollaries, which have not been discussed previously. Full article

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

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