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A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (25 November 2023) | Viewed by 15133

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Dr. Georgia Irina Oros

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Guest Editor

Department of Mathematics and Computer Science, Faculty of Informatics and Sciences, University of Oradea, 410087 Oradea, Romania
Interests: special classes of univalent functions; differential subordinations and superordinations; differential operators; integral operators; differential–integral operators
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue is a sequel to the successfully concluded first volume entitled “New Developments in Geometric Function Theory”. Following the same idea as the previous Special Issue, this new project aims to gather the latest developments in research concerning complex-valued functions from the Geometric Function Theory point of view.

Scholars’ contributions are expected on topics which include but are not limited to:

New classes of univalent and bi-univalent functions;
Studies regarding coefficient estimates including Fekete-Szegő functional, Hankel determinants, Toeplitz matrices;
Applications of different types of operators in Geometric Function Theory including differential, integral, fractional or quantum calculus operators;
Differential subordination and superordination theories in their classical form and also concerning their recent extensions, strong and fuzzy differential subordination and superordiantion theories;
Applications of different hypergeometric functions and orthogonal polynomials in the Geometric Function Theory.

The presentation of new results obtained by using any other techniques which can be applied in the field of complex analysis and its applications are welcome. Hopefully, new lines of research associated to the Geometric Function Theory are highlighted and will provide a boost in the development of this field.

Dr. Georgia Irina Oros
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

analytic function
univalent function
harmonic function
differential subordination
differential superordination
differential operator
integral operator
fractional operator
q-operator
special functions
orthogonal polynomials

Related Special Issues

New Developments in Geometric Function Theory in Axioms (15 articles)
New Developments in Geometric Function Theory III in Axioms (1 article)

Published Papers (15 papers)

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Editorial

Jump to: Research

4 pages, 152 KiB

Open AccessEditorial

New Developments in Geometric Function Theory II

by Georgia Irina Oros

Axioms 2024, 13(4), 224; https://doi.org/10.3390/axioms13040224 - 28 Mar 2024

Viewed by 732

Abstract

This Special Issue is a sequel to the successful first volume entitled “New Developments in Geometric Function Theory” [...] Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

Research

Jump to: Editorial

14 pages, 317 KiB

Open AccessArticle

Results of Third-Order Strong Differential Subordinations

by Madan Mohan Soren, Abbas Kareem Wanas and Luminiţa-Ioana Cotîrlǎ

Axioms 2024, 13(1), 42; https://doi.org/10.3390/axioms13010042 - 10 Jan 2024

Cited by 1 | Viewed by 933

Abstract

In this paper, we present and investigate the notion of third-order strong differential subordinations, unveiling several intriguing properties within the context of specific classes of admissible functions. Furthermore, we extend certain definitions, presenting novel and fascinating results. We also derive several interesting properties [...] Read more.

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

14 pages, 1021 KiB

Open AccessArticle

Coefficient Bounds for Some Families of Bi-Univalent Functions with Missing Coefficients

by Ebrahim Analouei Adegani, Mostafa Jafari, Teodor Bulboacă and Paweł Zaprawa

Axioms 2023, 12(12), 1071; https://doi.org/10.3390/axioms12121071 - 23 Nov 2023

Cited by 3 | Viewed by 900

Abstract

A branch of complex analysis with a rich history is geometric function theory, which first appeared in the early 20th century. The function theory deals with a variety of analytical tools to study the geometric features of complex-valued functions. The main purpose of [...] Read more.

| a_{n} |

of the functions that belong to a class of bi-univalent functions with missing coefficients that are defined by using the subordination. The significance of our present results consists of improvements to some previous results concerning different recent subclasses of bi-univalent functions, and the aim of this paper is to improve the results of previous outcomes. In addition, important examples of some classes of such functions are provided, which can help to understand the issues related to these functions. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

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Figure 1

13 pages, 337 KiB

Open AccessArticle

A New Pseudo-Type κ-Fold Symmetric Bi-Univalent Function Class

by Sondekola Rudra Swamy and Luminita-Ioana Cotîrlă

Axioms 2023, 12(10), 953; https://doi.org/10.3390/axioms12100953 - 10 Oct 2023

Cited by 1 | Viewed by 658

Abstract

We introduce and study a new pseudo-type

κ

-fold symmetric bi-univalent function class that meets certain subordination conditions in this article. For functions in the newly formed class, the initial coefficient bounds are obtained. For members in this class, the Fekete–Szegö issue is [...] Read more.

We introduce and study a new pseudo-type

κ

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

17 pages, 327 KiB

Open AccessArticle

Bi-Univalent Functions Based on Binomial Series-Type Convolution Operator Related with Telephone Numbers

by Hasan Bayram, Kaliappan Vijaya, Gangadharan Murugusundaramoorthy and Sibel Yalçın

Axioms 2023, 12(10), 951; https://doi.org/10.3390/axioms12100951 - 7 Oct 2023

Cited by 2 | Viewed by 868

Abstract

This paper introduces two novel subclasses of the function class

Σ

for bi-univalent functions, leveraging generalized telephone numbers and Binomial series through convolution. The exploration is conducted within the domain of the open unit disk. We delve into the analysis of initial Taylor-Maclaurin [...] Read more.

This paper introduces two novel subclasses of the function class

Σ

| a_{2} |

and

| a_{3} |

, deriving insights and findings for functions belonging to these new subclasses. Additionally, Fekete-Szegö inequalities are established for these functions. Furthermore, the study unveils a range of new subclasses of

Σ

, some of which are special cases, yet have not been previously explored in conjunction with telephone numbers. These subclasses emerge as a result of hybrid-type convolution operators. Concluding from our results, we present several corollaries, which stand as fresh contributions in the domain of involution numbers involving hybrid-type convolution operators. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

18 pages, 332 KiB

Open AccessArticle

Starlike Functions Associated with Bernoulli’s Numbers of Second Kind

by Mohsan Raza, Mehak Tariq, Jong-Suk Ro, Fairouz Tchier and Sarfraz Nawaz Malik

Axioms 2023, 12(8), 764; https://doi.org/10.3390/axioms12080764 - 3 Aug 2023

Cited by 1 | Viewed by 851

Abstract

The aim of this paper is to introduce a class of starlike functions that are related to Bernoulli’s numbers of the second kind. Let [...] Read more.

The aim of this paper is to introduce a class of starlike functions that are related to Bernoulli’s numbers of the second kind. Let

φ_{B S} (ξ) = {(\frac{ξ}{e^{ξ} - 1})}^{2} = \sum_{n = 0}^{\infty} \frac{ξ^{n} B_{n}^{2}}{n!}

, where the coefficients of

B_{n}^{2}

are Bernoulli numbers of the second kind. Then, we introduce a subclass of starlike functions

𝟊

such that

\frac{ξ 𝟊' (ξ)}{𝟊 (ξ)} ≺ φ_{B S} (ξ) .

We found out the coefficient bounds, several radii problems, structural formulas, and inclusion relations. We also found sharp Hankel determinant problems of this class. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

12 pages, 309 KiB

Open AccessArticle

Classes of Harmonic Functions Related to Mittag-Leffler Function

by Abeer A. Al-Dohiman, Basem Aref Frasin, Naci Taşar and Fethiye Müge Sakar

Axioms 2023, 12(7), 714; https://doi.org/10.3390/axioms12070714 - 23 Jul 2023

Cited by 2 | Viewed by 890

Abstract

The purpose of this paper is to find new inclusion relations of the harmonic class

HF (ϱ, γ)

with the subclasses

S_{HF}^{*},

K_{HF}

and

T N_{HF} (τ)

of harmonic functions by applying the [...] Read more.

The purpose of this paper is to find new inclusion relations of the harmonic class

HF (ϱ, γ)

with the subclasses

S_{HF}^{*},

K_{HF}

and

T N_{HF} (τ)

of harmonic functions by applying the convolution operator

Θ (ℑ)

associated with the Mittag-Leffler function. Further for

ϱ = 0

, several special cases of the main results are also obtained. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

19 pages, 352 KiB

Open AccessArticle

Faber Polynomial Coefficient Estimates for Bi-Close-to-Convex Functions Defined by the q-Fractional Derivative

by Hari Mohan Srivastava, Isra Al-Shbeil, Qin Xin, Fairouz Tchier, Shahid Khan and Sarfraz Nawaz Malik

Axioms 2023, 12(6), 585; https://doi.org/10.3390/axioms12060585 - 13 Jun 2023

Cited by 6 | Viewed by 1142

Abstract

By utilizing the concept of the q-fractional derivative operator and bi-close-to-convex functions, we define a new subclass of

A

, where the class

A

contains normalized analytic functions in the open unit disk

E

and is invariant or symmetric under rotation. First, [...] Read more.

By utilizing the concept of the q-fractional derivative operator and bi-close-to-convex functions, we define a new subclass of

A

, where the class

A

contains normalized analytic functions in the open unit disk

E

and is invariant or symmetric under rotation. First, using the Faber polynomial expansion (FPE) technique, we determine the lth coefficient bound for the functions contained within this class. We provide a further explanation for the first few coefficients of bi-close-to-convex functions defined by the q-fractional derivative. We also emphasize upon a few well-known outcomes of the major findings in this article. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

18 pages, 515 KiB

Open AccessArticle

Geometric Properties of Generalized Integral Operators Related to The Miller–Ross Function

by Sercan Kazımoğlu, Erhan Deniz and Luminita-Ioana Cotirla

Axioms 2023, 12(6), 563; https://doi.org/10.3390/axioms12060563 - 7 Jun 2023

Cited by 3 | Viewed by 846

Abstract

It is very well-known that the special functions and integral operators play a vital role in the research of applied and mathematical sciences. In this paper, our aim is to present sufficient conditions for the families of integral operators containing the normalized forms of the Miller–Ross functions such that they can be univalent in the open unit disk. Moreover, we find the convexity order of these operators. In proof of results, we use some differential inequalities related with Miller–Ross functions and well-known lemmas. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

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12 pages, 325 KiB

Open AccessArticle

Coefficient Estimation Utilizing the Faber Polynomial for a Subfamily of Bi-Univalent Functions

by Abdullah Alsoboh, Ala Amourah, Fethiye Müge Sakar, Osama Ogilat, Gharib Mousa Gharib and Nasser Zomot

Axioms 2023, 12(6), 512; https://doi.org/10.3390/axioms12060512 - 24 May 2023

Cited by 3 | Viewed by 1017

Abstract

The paper introduces a new family of analytic bi-univalent functions that are injective and possess analytic inverses, by employing a q-analogue of the derivative operator. Moreover, the article establishes the upper bounds of the Taylor–Maclaurin coefficients of these functions, which can aid in approximating the accuracy of approximations using a finite number of terms. The upper bounds are obtained by approximating analytic functions using Faber polynomial expansions. These bounds apply to both the initial few coefficients and all coefficients in the series, making them general and early, respectively. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

13 pages, 345 KiB

Open AccessArticle

Applications Laguerre Polynomials for Families of Bi-Univalent Functions Defined with (p,q)-Wanas Operator

by Abbas Kareem Wanas, Fethiye Müge Sakar and Alina Alb Lupaş

Axioms 2023, 12(5), 430; https://doi.org/10.3390/axioms12050430 - 26 Apr 2023

Cited by 4 | Viewed by 941

Abstract

In current manuscript, using Laguerre polynomials and

(p - q)

-Wanas operator, we identify upper bounds

|a_{2}|

and

|a_{3}|

which are first two Taylor-Maclaurin coefficients for a specific bi-univalent functions classes [...] Read more.

In current manuscript, using Laguerre polynomials and

(p - q)

-Wanas operator, we identify upper bounds

|a_{2}|

and

|a_{3}|

which are first two Taylor-Maclaurin coefficients for a specific bi-univalent functions classes

W_{\sum} (η, δ, λ, σ, θ, α, β, p, q; h)

and

K_{\sum} (ξ, ρ, σ, θ, α, β, p, q; h)

which cover the convex and starlike functions. Also, we discuss Fekete-Szegö type inequality for defined class. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

13 pages, 296 KiB

Open AccessArticle

A q-Analog of the Class of Completely Convex Functions and Lidstone Series

by Maryam Al-Towailb and Zeinab S. I. Mansour

Axioms 2023, 12(5), 412; https://doi.org/10.3390/axioms12050412 - 24 Apr 2023

Cited by 1 | Viewed by 853

Abstract

This paper introduces a q-analog of the class of completely convex functions. We prove specific properties, including that q-completely convex functions have convergent q-Lidstone series expansions. We also provide a sufficient and necessary condition for a real function to have [...] Read more.

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

13 pages, 316 KiB

Open AccessArticle

New Criteria for Convex-Exponent Product of Log-Harmonic Functions

by Rasoul Aghalary, Ali Ebadian, Nak Eun Cho and Mehri Alizadeh

Axioms 2023, 12(5), 409; https://doi.org/10.3390/axioms12050409 - 22 Apr 2023

Cited by 1 | Viewed by 860

Abstract

In this study, we consider different types of convex-exponent products of elements of a certain class of log-harmonic mapping and then find sufficient conditions for them to be starlike log-harmonic functions. For instance, we show that, if f is a spirallike function, then [...] Read more.

γ

, the log-harmonic mapping

F (z) = {f (z) | f (z) |}^{2 γ}

α

s p i r a l i k e

of order

ρ

. Our results generalize earlier work in the literature. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

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Figure 1

12 pages, 321 KiB

Open AccessArticle

Bell Distribution Series Defined on Subclasses of Bi-Univalent Functions That Are Subordinate to Horadam Polynomials

by Ibtisam Aldawish, Basem Frasin and Ala Amourah

Axioms 2023, 12(4), 362; https://doi.org/10.3390/axioms12040362 - 10 Apr 2023

Cited by 1 | Viewed by 893

Abstract

Several different subclasses of the bi-univalent function class

Σ

were introduced and studied by many authors using distribution series like Pascal distribution, Poisson distribution, Borel distribution, the Mittag-Leffler-type Borel distribution, Miller–Ross-Type Poisson Distribution. In the present paper, by making use of the Bell [...] Read more.

Several different subclasses of the bi-univalent function class

Σ

G_{Σ}^{t} (x, p, q, λ, β, γ)

of normalized bi-univalent functions in the open unit disk

U

, which are associated with the Horadam polynomials and estimate the second and the third coefficients in the Taylor-Maclaurin expansions of functions belonging to this class. Furthermore, we establish the Fekete–Szegö inequality for functions in the family

G_{Σ}^{t} (x, p, q, λ, β, γ) .

After specializing the parameters used in our main results, a number of new results are demonstrated to follow. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

11 pages, 293 KiB

Open AccessArticle

Coefficient Results concerning a New Class of Functions Associated with Gegenbauer Polynomials and Convolution in Terms of Subordination

by Sunday Olufemi Olatunji, Matthew Olanrewaju Oluwayemi and Georgia Irina Oros

Axioms 2023, 12(4), 360; https://doi.org/10.3390/axioms12040360 - 8 Apr 2023

Cited by 4 | Viewed by 851

Abstract

Gegenbauer polynomials constitute a numerical tool that has attracted the interest of many function theorists in recent times mainly due to their real-life applications in many areas of the sciences and engineering. Their applications in geometric function theory (GFT) have also been considered by many researchers. In this paper, this powerful tool is associated with the prolific concepts of convolution and subordination. The main purpose of the research contained in this paper is to introduce and study a new subclass of analytic functions. This subclass is presented using an operator defined as the convolution of the generalized distribution and the error function and applying the principle of subordination. Investigations into this subclass are considered in connection to Carathéodory functions, the modified sigmoid function and Bell numbers to obtain coefficient estimates for the contained functions. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

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