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Axioms
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New Developments in Geometric Function Theory
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New Developments in Geometric Function Theory

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

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 25040
Please contact the Guest Editor or the Journal Editor (alex.zhang@mdpi.com) for any queries about the scope, discount, submission procedure and publication process.

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Dr. Georgia Irina Oros

E-Mail Website
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 aims to gather the latest developments in the research concerning complex-valued functions from Geometric Function Theory point of view. Contributions are expected regarding any aspects of subordination and superordination results, different types of operators specific to the research in this field,  special functions connected to univalent functions’ theory. Hopefully, new approaches would emerge regarding the introduction and study of special classes of univalent functions using operators and the classical theories of differential subordination and superordination as well as the newer adapted theories of strong differential subordination and superordination and fuzzy differential subordination and superordination. Authors are invited to submit their latest results related to analytic functions in all their variety and also related to their applications in other fields of research. Quantum calculus and its applications in Geometric Function Theory is also expected to provide interesting outcome. Presentation of results obtained by using any other techniques which can be applied in the field of complex analysis and its applications are welcome.

This special issue devoted especially to complex analysis is proposed as a mean to find new approaches for Geometric Function Theory which to inspire further development of this field.

Prof. 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
  • special functions

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Published Papers (15 papers)

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Editorial

Jump to: Research

4 pages, 207 KiB  
Editorial
New Developments in Geometric Function Theory
by Georgia Irina Oros
Axioms 2023, 12(1), 59; https://doi.org/10.3390/axioms12010059 - 04 Jan 2023
Viewed by 1618
Abstract
This Special Issue aims to highlight the latest developments in the research concerning complex-valued functions from the perspective of geometric function theory [...] Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)

Research

Jump to: Editorial

13 pages, 272 KiB  
Article
Certain New Class of Analytic Functions Defined by Using a Fractional Derivative and Mittag-Leffler Functions
by Mohammad Faisal Khan, Shahid Khan, Saqib Hussain, Maslina Darus and Khaled Matarneh
Axioms 2022, 11(11), 655; https://doi.org/10.3390/axioms11110655 - 18 Nov 2022
Cited by 2 | Viewed by 1110
Abstract
Fractional calculus has a number of applications in the field of science, specially in mathematics. In this paper, we discuss some applications of fractional differential operators in the field of geometric function theory. Here, we combine the fractional differential operator and the Mittag-Leffler [...] Read more.
Fractional calculus has a number of applications in the field of science, specially in mathematics. In this paper, we discuss some applications of fractional differential operators in the field of geometric function theory. Here, we combine the fractional differential operator and the Mittag-Leffler functions to formulate and arrange a new operator of fractional calculus. We define a new class of normalized analytic functions by means of a newly defined fractional operator and discuss some of its interesting geometric properties in open unit disk. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
11 pages, 932 KiB  
Article
Geometric Study of 2D-Wave Equations in View of K-Symbol Airy Functions
by Samir B. Hadid and Rabha W. Ibrahim
Axioms 2022, 11(11), 590; https://doi.org/10.3390/axioms11110590 - 26 Oct 2022
Cited by 5 | Viewed by 1281
Abstract
The notion of k-symbol special functions has recently been introduced. This new concept offers many interesting geometric properties for these special functions including logarithmic convexity. The aim of the present paper is to exploit essentially two-dimensional wave propagation in the earth-ionosphere wave [...] Read more.
The notion of k-symbol special functions has recently been introduced. This new concept offers many interesting geometric properties for these special functions including logarithmic convexity. The aim of the present paper is to exploit essentially two-dimensional wave propagation in the earth-ionosphere wave path using k-symbol Airy functions (KAFs) in the open unit disk. It is shown that the standard wave-mode working formula may be determined by orthogonality considerations without the use of intricate justifications of the complex plane. By taking into account the symmetry-convex depiction of the KAFs, the formula combination is derived. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
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7 pages, 269 KiB  
Article
Applications of Beta Negative Binomial Distribution and Laguerre Polynomials on Ozaki Bi-Close-to-Convex Functions
by Isra Al-Shbeil, Abbas Kareem Wanas, Afis Saliu and Adriana Cătaş
Axioms 2022, 11(9), 451; https://doi.org/10.3390/axioms11090451 - 02 Sep 2022
Cited by 13 | Viewed by 1407
Abstract
In the present paper, due to beta negative binomial distribution series and Laguerre polynomials, we investigate a new family FΣ(δ,η,λ,θ;h) of normalized holomorphic and bi-univalent functions associated with Ozaki close-to-convex functions. [...] Read more.
In the present paper, due to beta negative binomial distribution series and Laguerre polynomials, we investigate a new family FΣ(δ,η,λ,θ;h) of normalized holomorphic and bi-univalent functions associated with Ozaki close-to-convex functions. We provide estimates on the initial Taylor–Maclaurin coefficients and discuss Fekete–Szegő type inequality for functions in this family. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
14 pages, 287 KiB  
Article
On Special Fuzzy Differential Subordinations Obtained for Riemann–Liouville Fractional Integral of Ruscheweyh and Sălăgean Operators
by Alina Alb Lupaş
Axioms 2022, 11(9), 428; https://doi.org/10.3390/axioms11090428 - 25 Aug 2022
Cited by 5 | Viewed by 958
Abstract
New results concerning fuzzy differential subordination theory are obtained in this paper using the operator denoted by Dz−λLαn, previously introduced by applying the Riemann–Liouville fractional integral to the convex combination of well-known Ruscheweyh and Sălăgean differential [...] Read more.
New results concerning fuzzy differential subordination theory are obtained in this paper using the operator denoted by Dz−λLαn, previously introduced by applying the Riemann–Liouville fractional integral to the convex combination of well-known Ruscheweyh and Sălăgean differential operators. A new fuzzy subclass DLnFδ,α,λ is defined and studied involving the operator Dz−λLαn. Fuzzy differential subordinations are obtained considering functions from class DLnFδ,α,λ and the fuzzy best dominants are also given. Using particular functions interesting corollaries are obtained and an example shows how the obtained results can be applied. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
14 pages, 323 KiB  
Article
Cauchy Integral and Boundary Value for Vector-Valued Tempered Distributions
by Richard D. Carmichael
Axioms 2022, 11(8), 392; https://doi.org/10.3390/axioms11080392 - 10 Aug 2022
Cited by 1 | Viewed by 1068
Abstract
Using the historically general growth condition on scalar-valued analytic functions, which have tempered distributions as boundary values, we show that vector-valued analytic functions in tubes TC=Rn+iC obtain vector-valued tempered distributions as boundary values. In a certain [...] Read more.
Using the historically general growth condition on scalar-valued analytic functions, which have tempered distributions as boundary values, we show that vector-valued analytic functions in tubes TC=Rn+iC obtain vector-valued tempered distributions as boundary values. In a certain vector-valued case, we study the structure of this boundary value, which is shown to be the Fourier transform of the distributional derivative of a vector-valued continuous function of polynomial growth. A set of vector-valued functions used to show the structure of the boundary value is shown to have a one–one and onto relationship with a set of vector-valued distributions, which generalize the Schwartz space DL2′(Rn); the tempered distribution Fourier transform defines the relationship between these two sets. By combining the previously stated results, we obtain a Cauchy integral representation of the vector-valued analytic functions in terms of the boundary value. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
10 pages, 276 KiB  
Article
New Results about Radius of Convexity and Uniform Convexity of Bessel Functions
by Luminiţa-Ioana Cotîrlă, Pál Aurel Kupán and Róbert Szász
Axioms 2022, 11(8), 380; https://doi.org/10.3390/axioms11080380 - 31 Jul 2022
Cited by 4 | Viewed by 1071
Abstract
We determine in this paper new results about the radius of uniform convexity of two kinds of normalization of the Bessel function Jν in the case ν∈(−2,−1), and provide an alternative proof regarding [...] Read more.
We determine in this paper new results about the radius of uniform convexity of two kinds of normalization of the Bessel function Jν in the case ν∈(−2,−1), and provide an alternative proof regarding the radius of convexity of order alpha. We then compare results regarding the convexity and uniform convexity of the considered functions and determine interesting connections between them. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
14 pages, 297 KiB  
Article
Sharp Bounds for the Second Hankel Determinant of Logarithmic Coefficients for Strongly Starlike and Strongly Convex Functions
by Sevtap Sümer Eker, Bilal Şeker, Bilal Çekiç and Mugur Acu
Axioms 2022, 11(8), 369; https://doi.org/10.3390/axioms11080369 - 28 Jul 2022
Cited by 8 | Viewed by 1426
Abstract
The logarithmic coefficients are very essential in the problems of univalent functions theory. The importance of the logarithmic coefficients is due to the fact that the bounds on logarithmic coefficients of f can transfer to the Taylor coefficients of univalent functions themselves or [...] Read more.
The logarithmic coefficients are very essential in the problems of univalent functions theory. The importance of the logarithmic coefficients is due to the fact that the bounds on logarithmic coefficients of f can transfer to the Taylor coefficients of univalent functions themselves or to their powers, via the Lebedev–Milin inequalities; therefore, it is interesting to investigate the Hankel determinant whose entries are logarithmic coefficients. The main purpose of this paper is to obtain the sharp bounds for the second Hankel determinant of logarithmic coefficients of strongly starlike functions and strongly convex functions. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
8 pages, 281 KiB  
Article
An Avant-Garde Construction for Subclasses of Analytic Bi-Univalent Functions
by Feras Yousef, Ala Amourah, Basem Aref Frasin and Teodor Bulboacă
Axioms 2022, 11(6), 267; https://doi.org/10.3390/axioms11060267 - 01 Jun 2022
Cited by 22 | Viewed by 1696
Abstract
The zero-truncated Poisson distribution is an important and appropriate model for many real-world applications. Here, we exploit the zero-truncated Poisson distribution probabilities to construct a new subclass of analytic bi-univalent functions involving Gegenbauer polynomials. For functions in the constructed class, we explore estimates [...] Read more.
The zero-truncated Poisson distribution is an important and appropriate model for many real-world applications. Here, we exploit the zero-truncated Poisson distribution probabilities to construct a new subclass of analytic bi-univalent functions involving Gegenbauer polynomials. For functions in the constructed class, we explore estimates of Taylor–Maclaurin coefficients a2 and a3, and next, we solve the Fekete–Szegő functional problem. A number of new interesting results are presented to follow upon specializing the parameters involved in our main results. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
13 pages, 364 KiB  
Article
Certain Subclasses of Bi-Starlike Function of Complex Order Defined by Erdély–Kober-Type Integral Operator
by Alhanouf Alburaikan, Gangadharan Murugusundaramoorthy and Sheza M. El-Deeb
Axioms 2022, 11(5), 237; https://doi.org/10.3390/axioms11050237 - 20 May 2022
Cited by 4 | Viewed by 1734
Abstract
In the present paper, we introduce new subclasses of bi-starlike and bi-convex functions of complex order associated with Erdély–Kober-type integral operator in the open unit disc and find the estimates of initial coefficients in these classes. Moreover, we obtain Fekete-Szegő inequalities for functions [...] Read more.
In the present paper, we introduce new subclasses of bi-starlike and bi-convex functions of complex order associated with Erdély–Kober-type integral operator in the open unit disc and find the estimates of initial coefficients in these classes. Moreover, we obtain Fekete-Szegő inequalities for functions in these classes. Some of the significances of our results are pointed out as corollaries. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
13 pages, 484 KiB  
Article
Applications of Confluent Hypergeometric Function in Strong Superordination Theory
by Georgia Irina Oros, Gheorghe Oros and Ancuța Maria Rus
Axioms 2022, 11(5), 209; https://doi.org/10.3390/axioms11050209 - 29 Apr 2022
Cited by 4 | Viewed by 1938
Abstract
In the research presented in this paper, confluent hypergeometric function is embedded in the theory of strong differential superordinations. In order to proceed with the study, the form of the confluent hypergeometric function is adapted taking into consideration certain classes of analytic functions [...] Read more.
In the research presented in this paper, confluent hypergeometric function is embedded in the theory of strong differential superordinations. In order to proceed with the study, the form of the confluent hypergeometric function is adapted taking into consideration certain classes of analytic functions depending on an extra parameter previously introduced related to the theory of strong differential subordination and superordination. Operators previously defined using confluent hypergeometric function, namely Kummer–Bernardi and Kummer–Libera integral operators, are also adapted to those classes and strong differential superordinations are obtained for which they are the best subordinants. Similar results are obtained regarding the derivatives of the operators. The examples presented at the end of the study are proof of the applicability of the original results. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
11 pages, 294 KiB  
Article
Hadamard Product Properties for Certain Subclasses of p-Valent Meromorphic Functions
by Alaa H. El-Qadeem and Ibrahim S. Elshazly
Axioms 2022, 11(4), 172; https://doi.org/10.3390/axioms11040172 - 13 Apr 2022
Cited by 3 | Viewed by 1489
Abstract
We study the Hadamard product features of certain subclasses of p-valent meromorphic functions defined in the punctured open-unit disc using the q-difference operator. For functions belonging to these subclasses, we obtained certain coefficient estimates and inclusion characteristics. Furthermore, linkages between the results [...] Read more.
We study the Hadamard product features of certain subclasses of p-valent meromorphic functions defined in the punctured open-unit disc using the q-difference operator. For functions belonging to these subclasses, we obtained certain coefficient estimates and inclusion characteristics. Furthermore, linkages between the results given here and those found in previous publications are highlighted. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
13 pages, 331 KiB  
Article
Subclasses of Yamakawa-Type Bi-Starlike Functions Associated with Gegenbauer Polynomials
by Gangadharan Murugusundaramoorthy and Teodor Bulboacă
Axioms 2022, 11(3), 92; https://doi.org/10.3390/axioms11030092 - 24 Feb 2022
Cited by 12 | Viewed by 1958
Abstract
In this paper, we introduce and investigate new subclasses (Yamakawa-type bi-starlike functions and another class of Lashin, both mentioned in the reference list) of bi-univalent functions defined in the open unit disk, which are associated with the Gegenbauer polynomials and satisfy subordination conditions. [...] Read more.
In this paper, we introduce and investigate new subclasses (Yamakawa-type bi-starlike functions and another class of Lashin, both mentioned in the reference list) of bi-univalent functions defined in the open unit disk, which are associated with the Gegenbauer polynomials and satisfy subordination conditions. Furthermore, we find estimates for the Taylor–Maclaurin coefficients |a2| and |a3| for functions in these new subclasses. Several known or new consequences of the results are also pointed out. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
10 pages, 261 KiB  
Article
An Application of Sălăgean Operator Concerning Starlike Functions
by Hatun Özlem Güney, Georgia Irina Oros and Shigeyoshi Owa
Axioms 2022, 11(2), 50; https://doi.org/10.3390/axioms11020050 - 27 Jan 2022
Cited by 8 | Viewed by 1877
Abstract
As an application of the well-known Sălăgean differential operator, a new operator is introduced and, using this, a new class of functions Sn(α) is defined, which has the classes of starlike and convex functions of order α as special [...] Read more.
As an application of the well-known Sălăgean differential operator, a new operator is introduced and, using this, a new class of functions Sn(α) is defined, which has the classes of starlike and convex functions of order α as special cases. Original results related to the newly defined class are obtained using the renowned Jack–Miller–Mocanu lemma. A relevant example is given regarding the applications of a new proven result concerning interesting properties of class Sn(α). Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
24 pages, 390 KiB  
Article
Generalized Vector-Valued Hardy Functions
by Richard D. Carmichael
Axioms 2022, 11(2), 39; https://doi.org/10.3390/axioms11020039 - 20 Jan 2022
Cited by 2 | Viewed by 1750
Abstract
We consider analytic functions in tubes Rn+iB⊂Cn with values in Banach space or Hilbert space. The base of the tube B will be a proper open connected subset of Rn, an open connected cone [...] Read more.
We consider analytic functions in tubes Rn+iB⊂Cn with values in Banach space or Hilbert space. The base of the tube B will be a proper open connected subset of Rn, an open connected cone in Rn, an open convex cone in Rn, and a regular cone in Rn, with this latter cone being an open convex cone which does not contain any entire straight lines. The analytic functions satisfy several different growth conditions in Lp norm, and all of the resulting spaces of analytic functions generalize the vector valued Hardy space Hp in Cn. The analytic functions are represented as the Fourier–Laplace transform of certain vector valued Lp functions which are characterized in the analysis. We give a characterization of the spaces of analytic functions in which the spaces are in fact subsets of the Hardy functions Hp. We obtain boundary value results on the distinguished boundary Rn+i{0¯} and on the topological boundary Rn+i∂B of the tube for the analytic functions in the Lp and vector valued tempered distribution topologies. Suggestions for associated future research are given. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
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