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Current Topics in Geometric Function Theory
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Current Topics in Geometric Function Theory

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: 31 July 2024 | Viewed by 13756

Special Issue Editor

Prof. Dr. Nicoleta Breaz

E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Computer Science and Engineering, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania
Interests: analytic functions; univalence; convexity; starlikeness; integral operators; regression modeling; smoothing spline

Special Issue Information

Dear Colleagues,  

One of the most studied branches in the theory of functions of one complex variable, concerned with the study of the geometric properties of analytical functions in complex analysis, the geometric theory of analytic functions (also called geometric function theory (GFT)), has in its core the Riemann mapping theorem, formulated by B. Riemann in 1851 and approached later by others such as C. Carathéodory, P. Koebe and L. Bieberbach. The duality of this field, based on the tradeoff between an analytical approach and geometric intuition, constitutes an advantage when we want to study the geometrical behavior of various classes of functions. The current development of the geometric function theory involving both classic and modern topics also generates many connections with various fields of mathematics, including special functions, probability distributions, fractional and q-calculus. Even if the geometric function theory is mostly viewed as a theoretical domain, significant practical applications were also obtained from the theoretical results in different fields, such as fluid mechanics, nuclear physics, mathematical physics, astrophysics and, more recently, in control theory, signal and image processing and others. 

This Special Issue aims to be a collection of original and recent research in the current topics of the field of geometric function theory related, but not restricted, to univalent function theory, study of starlike, convex and other classes of analytic functions with geometric properties, study of integral operators, differential subordination and superordination, and the newly flourishing research area based on q-calculus and fractional calculus. Research papers focusing on the geometric function theory used in real-life applications are also encouraged for this Special Issue.  

We are looking forward to receiving original contributions that can broaden the horizons of this research area. 

Prof. Dr. Nicoleta Breaz
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • classes of analytic functions
  • univalent functions
  • differential subordination and superordination
  • operator-related problems
  • quantum calculus
  • fractional calculus
  • extremal problems
  • preserving class properties
  • coefficients estimates
  • GFT in real-life applications

Published Papers (18 papers)

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Research

15 pages, 286 KiB  
Article
Generalized n-Polynomial p-Convexity and Related Inequalities
by Serap Özcan and Luminiţa-Ioana Cotîrlă
Mathematics 2024, 12(7), 1042; https://doi.org/10.3390/math12071042 - 30 Mar 2024
Viewed by 490
Abstract
In this paper, we construct a new class of convex functions, so-called generalized n-polynomial p-convex functions. We investigate their algebraic properties and provide some relationships between these functions and other types of convex functions. We establish Hermite–Hadamard (H–H) inequality for the [...] Read more.
In this paper, we construct a new class of convex functions, so-called generalized n-polynomial p-convex functions. We investigate their algebraic properties and provide some relationships between these functions and other types of convex functions. We establish Hermite–Hadamard (H–H) inequality for the newly defined class of functions. Additionally, we derive refinements of H–H inequality for functions whose first derivatives in absolute value at certain power are generalized n-polynomial p-convex. When p=−1, our definition evolves into a new definition for the class of convex functions so-called generalized n-polynomial harmonically convex functions. The results obtained in this study generalize regarding those found in the existing literature. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
13 pages, 415 KiB  
Article
Fekete–Szegő and Zalcman Functional Estimates for Subclasses of Alpha-Convex Functions Related to Trigonometric Functions
by Krishnan Marimuthu, Uma Jayaraman and Teodor Bulboacă
Mathematics 2024, 12(2), 234; https://doi.org/10.3390/math12020234 - 11 Jan 2024
Cited by 1 | Viewed by 602
Abstract
In this study, we introduce the new subclasses, Mα(sin) and Mα(cos), of α-convex functions associated with sine and cosine functions. First, we obtain the initial coefficient bounds for the first five coefficients of [...] Read more.
In this study, we introduce the new subclasses, Mα(sin) and Mα(cos), of α-convex functions associated with sine and cosine functions. First, we obtain the initial coefficient bounds for the first five coefficients of the functions that belong to these classes. Further, we determine the upper bound of the Zalcman functional for the class Mα(cos) for the case n=3, proving that the Zalcman conjecture holds for this value of n. Moreover, the problem of the Fekete–Szegő functional estimate for these classes is studied. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
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10 pages, 256 KiB  
Article
Bi-Univalency of m-Fold Symmetric Functions Associated with a Generalized Distribution
by Sunday Oluwafemi Olatunji, Fethiye Müge Sakar, Nicoleta Breaz, Seher Melike Aydoǧan and Matthew Olanrewaju Oluwayemi
Mathematics 2024, 12(2), 169; https://doi.org/10.3390/math12020169 - 5 Jan 2024
Viewed by 608
Abstract
The m-fold symmetric in terms of a generalized distribution series has not been considered in the literature. In this study, however, the authors investigated the bi-univalency of m-fold symmetric functions for the generalized distribution of two subclasses of analytic functions. The [...] Read more.
The m-fold symmetric in terms of a generalized distribution series has not been considered in the literature. In this study, however, the authors investigated the bi-univalency of m-fold symmetric functions for the generalized distribution of two subclasses of analytic functions. The initial few coefficient bounds amS and a2mS are obtained for the two subclasses of functions defined and the results serve as a new generalization in this direction. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
15 pages, 336 KiB  
Article
Certain Quantum Operator Related to Generalized Mittag–Leffler Function
by Mansour F. Yassen and Adel A. Attiya
Mathematics 2023, 11(24), 4963; https://doi.org/10.3390/math11244963 - 15 Dec 2023
Cited by 1 | Viewed by 650
Abstract
In this paper, we present a novel class of analytic functions in the form h(z)=zp+∑k=p+1∞akzk in the unit disk. These functions establish a connection between [...] Read more.
In this paper, we present a novel class of analytic functions in the form h(z)=zp+∑k=p+1∞akzk in the unit disk. These functions establish a connection between the extended Mittag–Leffler function and the quantum operator presented in this paper, which is denoted by ℵq,pn(L,a,b) and is also an extension of the Raina function that combines with the Jackson derivative. Through the application of differential subordination methods, essential properties like bounds of coefficients and the Fekete–Szegő problem for this class are derived. Additionally, some results of special cases to this study that were previously studied were also highlighted. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
10 pages, 285 KiB  
Article
Coefficient Estimates for Quasi-Subordination Classes Connected with the Combination of q-Convolution and Error Function
by Sheza M. El-Deeb and Luminita-Ioana Cotîrlă
Mathematics 2023, 11(23), 4834; https://doi.org/10.3390/math11234834 - 30 Nov 2023
Viewed by 604
Abstract
We utilize quasi-subordination to analyze and introduce several new classes, and we construct a new operator by combining the error function and q-convolution. Additionally, we obtain estimates for the Fekete Szego functional and the Taylor–Maclaurin coefficients for functions in c2 and [...] Read more.
We utilize quasi-subordination to analyze and introduce several new classes, and we construct a new operator by combining the error function and q-convolution. Additionally, we obtain estimates for the Fekete Szego functional and the Taylor–Maclaurin coefficients for functions in c2 and c3 new classes. Moreover, we discuss some applications of the operator. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
11 pages, 284 KiB  
Article
An Application for Bi-Concave Functions Associated with q-Convolution
by Sheza M. El-Deeb and Adriana Catas
Mathematics 2023, 11(22), 4680; https://doi.org/10.3390/math11224680 - 17 Nov 2023
Viewed by 583
Abstract
The aim of this paper is to introduce and investigate some new subclasses of bi-concave functions using q-convolution and some applications. These special cases are obtaining by making use of a q- derivative linear operator. For the new introduced subclasses, the [...] Read more.
The aim of this paper is to introduce and investigate some new subclasses of bi-concave functions using q-convolution and some applications. These special cases are obtaining by making use of a q- derivative linear operator. For the new introduced subclasses, the authors obtain the first two initial Taylor–Maclaurin coefficients |c2| and |c3| of bi-concave functions. For certain values of the parameters, the authors deduce interesting corollaries for coefficient bounds which imply special cases of the new introduced operator. Also, we develop two examples for coefficients |c2| and |c3| for certain functions. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
17 pages, 297 KiB  
Article
Fuzzy Differential Subordination Associated with a General Linear Transformation
by Sarfraz Nawaz Malik, Nazar Khan, Ferdous M. O. Tawfiq, Mohammad Faisal Khan, Qazi Zahoor Ahmad and Qin Xin
Mathematics 2023, 11(22), 4582; https://doi.org/10.3390/math11224582 - 8 Nov 2023
Viewed by 586
Abstract
In this study, we investigate a possible relationship between fuzzy differential subordination and the theory of geometric functions. First, using the Al-Oboudi differential operator and the Babalola convolution operator, we establish the new operator BSα,λm,t:A [...] Read more.
In this study, we investigate a possible relationship between fuzzy differential subordination and the theory of geometric functions. First, using the Al-Oboudi differential operator and the Babalola convolution operator, we establish the new operator BSα,λm,t:An→An in the open unit disc U. The second step is to develop fuzzy differential subordination for the operator BSα,λm,t. By considering linear transformations of the operator BSα,λm,t, we define a new fuzzy class of analytic functions in U which we denote by Tϝλ,t(m,α,δ). Several innovative results are found using the concept of fuzzy differential subordination and the operator BSα,λm,t for the function f in the class Tϝλ,t(m,α,δ). In addition, we explore a number of examples and corollaries to illustrate the implications of our key findings. Finally, we highlight several established results to demonstrate the connections between our work and existing studies. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
17 pages, 334 KiB  
Article
Concerning a Novel Integral Operator and a Specific Category of Starlike Functions
by Ayotunde Olajide Lasode, Timothy Oloyede Opoola, Isra Al-Shbeil, Timilehin Gideon Shaba and Huda Alsaud
Mathematics 2023, 11(21), 4519; https://doi.org/10.3390/math11214519 - 2 Nov 2023
Cited by 1 | Viewed by 728
Abstract
In this study, a novel integral operator that extends the functionality of some existing integral operators is presented. Specifically, the integral operator acts as the inverse operator to the widely recognized Opoola differential operator. By making use of the integral operator, a certain [...] Read more.
In this study, a novel integral operator that extends the functionality of some existing integral operators is presented. Specifically, the integral operator acts as the inverse operator to the widely recognized Opoola differential operator. By making use of the integral operator, a certain subclass of analytic univalent functions defined in the unit disk is proposed and investigated. This new class encompasses some familiar subclasses, like the class of starlike and the class of convex functions, while some new ones are introduced. The investigation thereafter covers coefficient inequality, distortion, growth, covering, integral preserving, closure, subordinating factor sequence, and integral means properties. Furthermore, the radii problems associated with this class are successfully addressed. Additionally, a few remarks are provided, to show that the novel integral operator and the new class generalize some existing ones. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
15 pages, 322 KiB  
Article
Strong Differential Subordinations and Superordinations for Riemann–Liouville Fractional Integral of Extended q-Hypergeometric Function
by Alina Alb Lupaş and Georgia Irina Oros
Mathematics 2023, 11(21), 4474; https://doi.org/10.3390/math11214474 - 28 Oct 2023
Viewed by 702
Abstract
The notions of strong differential subordination and its dual, strong differential superordination, have been introduced as extensions of the classical differential subordination and superordination concepts, respectively. The dual theories have developed nicely, and important results have been obtained involving different types of operators [...] Read more.
The notions of strong differential subordination and its dual, strong differential superordination, have been introduced as extensions of the classical differential subordination and superordination concepts, respectively. The dual theories have developed nicely, and important results have been obtained involving different types of operators and certain hypergeometric functions. In this paper, quantum calculus and fractional calculus aspects are added to the study. The well-known q-hypergeometric function is given a form extended to fit the study concerning previously introduced classes of functions specific to strong differential subordination and superordination theories. Riemann–Liouville fractional integral of extended q-hypergeometric function is defined here, and it is involved in the investigation of strong differential subordinations and superordinations. The best dominants and the best subordinants are provided in the theorems that are proved for the strong differential subordinations and superordinations, respectively. For particular functions considered due to their remarkable geometric properties as best dominant or best subordinant, interesting corollaries are stated. The study is concluded by connecting the results obtained using the dual theories through sandwich-type theorems and corollaries. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
16 pages, 340 KiB  
Article
On a New Class of Bi-Close-to-Convex Functions with Bounded Boundary Rotation
by Daniel Breaz, Prathviraj Sharma, Srikandan Sivasubramanian and Sheza M. El-Deeb
Mathematics 2023, 11(20), 4376; https://doi.org/10.3390/math11204376 - 21 Oct 2023
Cited by 2 | Viewed by 838
Abstract
In the current article, we introduce a new class of bi-close-to-convex functions with bounded boundary rotation. For this new class, the authors obtain the first three initial coefficient bounds of the newly defined bi-close-to-convex functions with bounded boundary rotation. By choosing special bi-convex [...] Read more.
In the current article, we introduce a new class of bi-close-to-convex functions with bounded boundary rotation. For this new class, the authors obtain the first three initial coefficient bounds of the newly defined bi-close-to-convex functions with bounded boundary rotation. By choosing special bi-convex functions, the authors obtain the first three initial coefficient bounds in the last section. The authors also verify the special cases where the familiar Brannan and Clunie’s conjecture is satisfied. Furthermore, the famous Fekete–Szegö inequality is also obtained for this new class of functions. Apart from the new interesting results, some of the results presented here improves the earlier results existing in the literature. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
10 pages, 286 KiB  
Article
Properties of a Special Holomorphic Function Linked with a Generalized Multiplier Transformation
by Sondekola Rudra Swamy, Alina Alb Lupaş, Nanjundan Magesh and Yerragunta Sailaja
Mathematics 2023, 11(19), 4126; https://doi.org/10.3390/math11194126 - 29 Sep 2023
Viewed by 553
Abstract
In the present paper, we introduce a special holomorphic function in U={z∈C:|z|<1} which is associated with new generalized multiplier transformations. We investigate several properties of the defined function using the concept [...] Read more.
In the present paper, we introduce a special holomorphic function in U={z∈C:|z|<1} which is associated with new generalized multiplier transformations. We investigate several properties of the defined function using the concept of subordination, then highlight a number of cases with interesting results. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
14 pages, 322 KiB  
Article
Some Results on Third-Order Differential Subordination and Differential Superordination for Analytic Functions Using a Fractional Differential Operator
by Faten Fakher Abdulnabi, Hiba F. Al-Janaby, Firas Ghanim and Alina Alb Lupaș
Mathematics 2023, 11(18), 4021; https://doi.org/10.3390/math11184021 - 21 Sep 2023
Cited by 1 | Viewed by 944
Abstract
In this study, we explore the implications of a third-order differential subordination in the context of analytic functions associated with fractional differential operators. Our investigation involves the consideration of specific admissible classes of third-order differential functions. We also extend this exploration to establish [...] Read more.
In this study, we explore the implications of a third-order differential subordination in the context of analytic functions associated with fractional differential operators. Our investigation involves the consideration of specific admissible classes of third-order differential functions. We also extend this exploration to establish a dual principle, resulting in a sandwich-type outcome. We introduce these admissible function classes by employing the fractional derivative operator DzαSN,Sϑz  and derive conditions on the normalized analytic function f that lead to sandwich-type subordination in combination with an appropriate fractional differential operator. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
10 pages, 788 KiB  
Article
Certain Results on Fuzzy p-Valent Functions Involving the Linear Operator
by Ekram Elsayed Ali, Miguel Vivas-Cortez, Shujaat Ali Shah and Abeer M. Albalahi
Mathematics 2023, 11(18), 3968; https://doi.org/10.3390/math11183968 - 19 Sep 2023
Cited by 1 | Viewed by 647
Abstract
The idea of fuzzy differential subordination is a generalisation of the traditional idea of differential subordination that evolved in recent years as a result of incorporating the idea of fuzzy set into the field of geometric function theory. In this investigation, we define [...] Read more.
The idea of fuzzy differential subordination is a generalisation of the traditional idea of differential subordination that evolved in recent years as a result of incorporating the idea of fuzzy set into the field of geometric function theory. In this investigation, we define some general classes of p-valent analytic functions defined by the fuzzy subordination and generalizes the various classical results of the multivalent functions. Our main focus is to define fuzzy multivalent functions and discuss some interesting inclusion results and various other useful properties of some subclasses of fuzzy p-valent functions, which are defined here by means of a certain generalized Srivastava-Attiya operator. Additionally, links between the significant findings of this study and preceding ones are also pointed out. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
15 pages, 303 KiB  
Article
Bounds for Toeplitz Determinants and Related Inequalities for a New Subclass of Analytic Functions
by Huo Tang, Ihtesham Gul, Saqib Hussain and Saima Noor
Mathematics 2023, 11(18), 3966; https://doi.org/10.3390/math11183966 - 18 Sep 2023
Cited by 1 | Viewed by 738
Abstract
In this article, we use the q-derivative operator and the principle of subordination to define a new subclass of analytic functions related to the q-Ruscheweyh operator. Sufficient conditions, sharp bounds for the initial coefficients, a Fekete–Szegö functional and a Toeplitz determinant [...] Read more.
In this article, we use the q-derivative operator and the principle of subordination to define a new subclass of analytic functions related to the q-Ruscheweyh operator. Sufficient conditions, sharp bounds for the initial coefficients, a Fekete–Szegö functional and a Toeplitz determinant are investigated for this new class of functions. Additionally, we also present several established consequences derived from our primary findings. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
20 pages, 336 KiB  
Article
New Criteria for Starlikness and Convexity of a Certain Family of Integral Operators
by Hari M. Srivastava, Rogayeh Alavi, Saeid Shams, Rasoul Aghalary and Santosh B. Joshi
Mathematics 2023, 11(18), 3919; https://doi.org/10.3390/math11183919 - 14 Sep 2023
Cited by 1 | Viewed by 663
Abstract
In this paper, we first modify one of the most famous theorems on the principle of differential subordination to hold true for normalized analytic functions with a fixed initial Taylor-Maclaurin coefficient. By using this modified form, we generalize and improve several results, which [...] Read more.
In this paper, we first modify one of the most famous theorems on the principle of differential subordination to hold true for normalized analytic functions with a fixed initial Taylor-Maclaurin coefficient. By using this modified form, we generalize and improve several results, which appeared recently in the literature on the geometric function theory of complex analysis. We also prove some simple conditions for starlikeness, convexity, and the strong starlikeness of several one-parameter families of integral operators, including (for example) a certain μ-convex integral operator and the familiar Bernardi integral operator. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
21 pages, 457 KiB  
Article
On Some Classes of Harmonic Functions Associated with the Janowski Function
by Lina Ma, Shuhai Li and Huo Tang
Mathematics 2023, 11(17), 3666; https://doi.org/10.3390/math11173666 - 25 Aug 2023
Viewed by 1095
Abstract
In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, we discuss the geometric properties of the classes, such [...] Read more.
In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, we discuss the geometric properties of the classes, such as the integral expression, coefficient estimation, distortion theorem, Jacobian estimation, growth estimates, and covering theorem. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
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22 pages, 357 KiB  
Article
Investigation of the Hankel Determinant Sharp Bounds for a Specific Analytic Function Linked to a Cardioid-Shaped Domain
by Isra Al-Shbeil, Muhammad Imran Faisal, Muhammad Arif, Muhammad Abbas and Reem K. Alhefthi
Mathematics 2023, 11(17), 3664; https://doi.org/10.3390/math11173664 - 25 Aug 2023
Viewed by 792
Abstract
One of the challenging tasks in the study of function theory is how to obtain sharp estimates of coefficients that appear in the Taylor–Maclaurin series of analytic univalent functions, and for obtaining these bounds, researchers used the concepts of Carathéodory functions. Among these [...] Read more.
One of the challenging tasks in the study of function theory is how to obtain sharp estimates of coefficients that appear in the Taylor–Maclaurin series of analytic univalent functions, and for obtaining these bounds, researchers used the concepts of Carathéodory functions. Among these coefficient-related problems, the problem of the third-order Hankel determinant sharp bound is the most difficult one. The aim of the present study is to determine the sharp bound of the Hankel determinant of third order by using the methodology of the aforementioned Carathéodory function family. Further, we also study some other coefficient-related problems, such as the Fekete–Szegő inequality and the second-order Hankel determinant. We examine these results for the family of bounded turning functions linked with a cardioid-shaped domain. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
20 pages, 325 KiB  
Article
Fuzzy Differential Subordination and Superordination Results for Fractional Integral Associated with Dziok-Srivastava Operator
by Alina Alb Lupaş
Mathematics 2023, 11(14), 3129; https://doi.org/10.3390/math11143129 - 15 Jul 2023
Cited by 2 | Viewed by 592
Abstract
Fuzzy set theory, introduced by Zadeh, gives an adaptable and logical solution to the provocation of introducing, evaluating, and opposing numerous sustainability scenarios. The results described in this article use the fuzzy set concept embedded into the theories of differential subordination and superordination [...] Read more.
Fuzzy set theory, introduced by Zadeh, gives an adaptable and logical solution to the provocation of introducing, evaluating, and opposing numerous sustainability scenarios. The results described in this article use the fuzzy set concept embedded into the theories of differential subordination and superordination from the geometric function theory. In 2011, fuzzy differential subordination was defined as an extension of the classical notion of differential subordination, and in 2017, the dual concept of fuzzy differential superordination appeared. These dual notions are applied in this paper regarding the fractional integral applied to Dziok–Srivastava operator. New fuzzy differential subordinations are proved using known lemmas, and the fuzzy best dominants are established for the obtained fuzzy differential subordinations. Dual results regarding fuzzy differential superordinations are proved for which the fuzzy best subordinates are shown. These are the first results that link the fractional integral applied to Dziok–Srivastava operator to fuzzy theory. Full article
(This article belongs to the Special Issue Current Topics in Geometric Function Theory)
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