Symmetry in Geometric Functions and Mathematical Analysis II

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 13566

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Department of Mathematical Analysis, Uniwersytet Rzeszowski, al. Rejtana 16c, 35-959 Rzeszów, Poland
Interests: complex analysis; geometric functions theory; differential subordinations
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Special Issue Information

Dear Colleagues,

This Special Issue, entitled “Symmetry in Geometric Functions and Mathematical Analysis”, is devoted to the publication of high-quality research, especially relating to geometrical aspects and harmonic and quasiconformal mappings (including applications in allied areas of mathematics and mathematical sciences). The issue will provide a forum for researchers and scientists to communicate their recent developments and to present recent results in the complex analysis theory of one and several variables, as well as application in algebraic geometry, number theory, and in physics, including the branches of hydrodynamics and quantum mechanics.

The research topics include but are not limited to:

  1. Complex analysis and potential theory;
  2. Partial differential equations;
  3. Geometrical aspects of complex analysis;
  4. Complex approximation theory;
  5. Harmonic and quasiconformal mappings;
  6. Generalized complex analysis;
  7. Complex dynamical systems and fractals;
  8. Entire and meromorphic functions;
  9. Applications.

Prof. Dr. Stanisława Kanas
Guest Editor

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Keywords

  • harmonic and quasiconformal mappings
  • entire and meromorphic functions
  • univalent and multivalent functions
  • subordinations and complex operator theory
  • geometrical aspects of complex analysis
  • special functions
  • applications of symmetry in mathematical analysis

Published Papers (11 papers)

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Research

17 pages, 325 KiB  
Article
Some Analysis of the Coefficient-Related Problems for Functions of Bounded Turning Associated with a Symmetric Image Domain
by Muhammad Arif, Muhammad Abbas, Reem K. Alhefthi, Daniel Breaz, Luminiţa-Ioana Cotîrlă and Eleonora Rapeanu
Symmetry 2023, 15(11), 2090; https://doi.org/10.3390/sym15112090 - 20 Nov 2023
Viewed by 695
Abstract
In the last few years, numerous subfamilies of univalent functions, whether directly or indirectly associated with exponential functions, have been introduced and thoroughly investigated. Among these, the families Se*, Ce and Re defined by subordination to ez [...] Read more.
In the last few years, numerous subfamilies of univalent functions, whether directly or indirectly associated with exponential functions, have been introduced and thoroughly investigated. Among these, the families Se*, Ce and Re defined by subordination to ez have been intensively investigated. While the coefficient problem on the class Se* and Ce has been solved in many cases, in this paper, we mainly intend to compute the sharp estimates of some initial coefficients, the Feketo–Szegö inequality, and the sharp bounds of second- and third-order Hankel determinants for functions belonging to the class Re. This work has the potential to significantly enrich and enhance the exploration of univalent functions in conjunction with exponential functions, making the field more comprehensive and robust. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
9 pages, 492 KiB  
Article
The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry
by Abraham A. Ungar
Symmetry 2023, 15(8), 1487; https://doi.org/10.3390/sym15081487 - 27 Jul 2023
Viewed by 696
Abstract
Ptolemy’s theorem in Euclidean geometry, named after the Greek astronomer and mathematician Claudius Ptolemy, is well known. We translate Ptolemy’s theorem from analytic Euclidean geometry into the Poincaré ball model of analytic hyperbolic geometry, which is based on the Möbius addition and its [...] Read more.
Ptolemy’s theorem in Euclidean geometry, named after the Greek astronomer and mathematician Claudius Ptolemy, is well known. We translate Ptolemy’s theorem from analytic Euclidean geometry into the Poincaré ball model of analytic hyperbolic geometry, which is based on the Möbius addition and its associated symmetry gyrogroup. The translation of Ptolemy’s theorem from Euclidean geometry into hyperbolic geometry is achieved by means of hyperbolic trigonometry, called gyrotrigonometry, to which the Poincaré ball model gives rise, and by means of the duality of trigonometry and gyrotrigonometry. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
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15 pages, 301 KiB  
Article
Applications of the Symmetric Quantum-Difference Operator for New Subclasses of Meromorphic Functions
by Isra Al-shbeil, Shahid Khan, Hala AlAqad, Salam Alnabulsi and Mohammad Faisal Khan
Symmetry 2023, 15(7), 1439; https://doi.org/10.3390/sym15071439 - 18 Jul 2023
Cited by 2 | Viewed by 730
Abstract
Our goal in this article is to use ideas from symmetric q-calculus operator theory in the study of meromorphic functions on the punctured unit disc and to propose a novel symmetric q-difference operator for these functions. A few additional classes of [...] Read more.
Our goal in this article is to use ideas from symmetric q-calculus operator theory in the study of meromorphic functions on the punctured unit disc and to propose a novel symmetric q-difference operator for these functions. A few additional classes of meromorphic functions are then defined in light of this new symmetric q-difference operator. We prove many useful conclusions regarding these newly constructed classes of meromorphic functions, such as convolution, subordination features, integral representations, and necessary conditions. The technique presented in this article may be used to produce a wide variety of new types of generalized symmetric q-difference operators, which can subsequently be used to investigate a wide variety of new classes of analytic and meromorphic functions related to symmetric quantum calculus. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
15 pages, 1907 KiB  
Article
Geometric Shape Characterisation Based on a Multi-Sweeping Paradigm
by Borut Žalik , Damjan Strnad , David Podgorelec , Ivana Kolingerová , Andrej Nerat , Niko Lukač , Štefan Kohek  and Luka Lukač 
Symmetry 2023, 15(6), 1212; https://doi.org/10.3390/sym15061212 - 6 Jun 2023
Cited by 1 | Viewed by 1075
Abstract
The characterisation of geometric shapes produces their concise description and is, therefore, important for subsequent analyses, for example in Computer Vision, Machine Learning, or shape matching. A new method for extracting characterisation vectors of 2D geometric shapes is proposed in this paper. The [...] Read more.
The characterisation of geometric shapes produces their concise description and is, therefore, important for subsequent analyses, for example in Computer Vision, Machine Learning, or shape matching. A new method for extracting characterisation vectors of 2D geometric shapes is proposed in this paper. The shape of interest, embedded into a raster space, is swept several times by sweep-lines having different slopes. The interior shape’s points, being in the middle of its boundary and laying on the actual sweep-line, are identified at each stage of the sweeping process. The midpoints are then connected iteratively into chains. The chains are filtered, vectorised, and normalised. The obtained polylines from the vectorisation step are used to design the shape’s characterisation vector for further application-specific analyses. The proposed method was verified on numerous shapes, where single- and multi-threaded implementations were compared. Finally, characterisation vectors, among which some were rotated and scaled, were determined for these shapes. The proposed method demonstrated a good rotation- and scaling-invariant identification of equal shapes. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
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22 pages, 351 KiB  
Article
New Applications of the Sălăgean Quantum Differential Operator for New Subclasses of q-Starlike and q-Convex Functions Associated with the Cardioid Domain
by Suha B. Al-Shaikh
Symmetry 2023, 15(6), 1185; https://doi.org/10.3390/sym15061185 - 1 Jun 2023
Cited by 1 | Viewed by 855
Abstract
In this paper, we define a new family of q-starlike and q-convex functions related to the cardioid domain utilizing the ideas of subordination and the Sălăgean quantum differential operator. The primary contribution of this article is the derivation of a sharp [...] Read more.
In this paper, we define a new family of q-starlike and q-convex functions related to the cardioid domain utilizing the ideas of subordination and the Sălăgean quantum differential operator. The primary contribution of this article is the derivation of a sharp inequality for the newly established subclasses of q-starlike and q-convex functions in the open unit disc U. For this novel family, bounds of the first two Taylor-Maclaurin coefficients, the Fekete-Szegö-type functional, and coefficient inequalities are studied. Furthermore, we also investigate some new results for the inverse function belonging to the classes of q-starlike and q-convex functions. The results presented in this article are sharp. To draw connections between the early and present findings, several well-known corollaries are also highlighted. Symmetric quantum calculus operator theory can be used to investigate the symmetry properties of this new family of functions. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
15 pages, 304 KiB  
Article
Second Natural Connection on Riemannian Π-Manifolds
by Hristo Manev
Symmetry 2023, 15(4), 817; https://doi.org/10.3390/sym15040817 - 28 Mar 2023
Cited by 1 | Viewed by 715
Abstract
An object of investigation is the differential geometry of the Riemannian Π-manifolds; in particular, a natural connection, determined by a property of its torsion tensor, is defined, and it is called the second natural connection on Riemannian Π-manifold. The uniqueness of [...] Read more.
An object of investigation is the differential geometry of the Riemannian Π-manifolds; in particular, a natural connection, determined by a property of its torsion tensor, is defined, and it is called the second natural connection on Riemannian Π-manifold. The uniqueness of this connection is proved, and a necessary and sufficient condition for coincidence with the known first natural connection on the considered manifolds is found. The form of the torsion tensor of the second natural connection is obtained in the classes of the Riemannian Π-manifolds, in which it differs from the first natural connection. All of the main classes of considered manifolds are characterized with respect to the torsion of the second natural connection. An explicit example of dimension 5 is given in support of the proven assertions. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
18 pages, 319 KiB  
Article
Sharp Coefficient Bounds for a New Subclass of q-Starlike Functions Associated with q-Analogue of the Hyperbolic Tangent Function
by Chetan Swarup
Symmetry 2023, 15(3), 763; https://doi.org/10.3390/sym15030763 - 20 Mar 2023
Cited by 5 | Viewed by 1125
Abstract
In this study, by making the use of q-analogous of the hyperbolic tangent function and a Sălăgean q-differential operator, a new class of q-starlike functions is introduced. The prime contribution of this study covers the derivation of sharp coefficient bounds [...] Read more.
In this study, by making the use of q-analogous of the hyperbolic tangent function and a Sălăgean q-differential operator, a new class of q-starlike functions is introduced. The prime contribution of this study covers the derivation of sharp coefficient bounds in open unit disk U, especially the first three coefficient bounds, Fekete–Szego type functional, and upper bounds of second- and third-order Hankel determinant for the functions to this class. We also use Zalcman and generalized Zalcman conjectures to investigate the coefficient bounds of a newly defined class of functions. Furthermore, some known corollaries are highlighted based on the unique choices of the involved parameters l and q. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
16 pages, 308 KiB  
Article
Properties of q-Symmetric Starlike Functions of Janowski Type
by Afis Saliu, Isra Al-Shbeil, Jianhua Gong, Sarfraz Nawaz Malik and Najla Aloraini
Symmetry 2022, 14(9), 1907; https://doi.org/10.3390/sym14091907 - 12 Sep 2022
Cited by 19 | Viewed by 1210
Abstract
The word “symmetry” is a Greek word that originated from “symmetria”. It means an agreement in dimensions, due proportion, and arrangement; however, in complex analysis, it means objects remaining invariant under some transformation. This idea has now been recently used in geometric function [...] Read more.
The word “symmetry” is a Greek word that originated from “symmetria”. It means an agreement in dimensions, due proportion, and arrangement; however, in complex analysis, it means objects remaining invariant under some transformation. This idea has now been recently used in geometric function theory to modify the earlier classical q-derivative introduced by Ismail et al. due to its better convergence properties. Consequently, we introduce a new class of analytic functions by using the notion of q-symmetric derivative. The investigation in this paper obtains a number of the latest important results in q-theory, including coefficient inequalities and convolution characterization of q-symmetric starlike functions related to Janowski mappings. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
14 pages, 293 KiB  
Article
Properties of a Subclass of Analytic Functions Defined by Using an Atangana–Baleanu Fractional Integral Operator
by Alina Alb Lupaş and Adriana Cătaş
Symmetry 2022, 14(4), 649; https://doi.org/10.3390/sym14040649 - 23 Mar 2022
Cited by 1 | Viewed by 1159
Abstract
The Atangana–Baleanu fractional integral and multiplier transformations are two functions successfully used separately in many recently published studies. They were previously combined and the resulting function was applied for obtaining interesting new results concerning the theories of differential subordination and fuzzy differential subordination. [...] Read more.
The Atangana–Baleanu fractional integral and multiplier transformations are two functions successfully used separately in many recently published studies. They were previously combined and the resulting function was applied for obtaining interesting new results concerning the theories of differential subordination and fuzzy differential subordination. In the present investigation, a new approach is taken by using the operator previously introduced by applying the Atangana–Baleanu fractional integral to a multiplier transformation for introducing a new subclass of analytic functions. Using the methods familiar to geometric function theory, certain geometrical properties of the newly introduced class are obtained such as coefficient estimates, distortion theorems, closure theorems, neighborhoods and the radii of starlikeness, convexity, and close-to-convexity of functions belonging to the class. This class may have symmetric or assymetric properties. The results could prove interesting for future studies due to the new applications of the operator and because the univalence properties of the new subclass of functions could inspire further investigations having it as the main focus. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
12 pages, 299 KiB  
Article
Fractional Integral of a Confluent Hypergeometric Function Applied to Defining a New Class of Analytic Functions
by Alina Alb Lupaş and Georgia Irina Oros
Symmetry 2022, 14(2), 427; https://doi.org/10.3390/sym14020427 - 21 Feb 2022
Cited by 3 | Viewed by 1210
Abstract
The study on fractional integrals of confluent hypergeometric functions provides interesting subordination and superordination results and inspired the idea of using this operator to introduce a new class of analytic functions. Given the class of functions [...] Read more.
The study on fractional integrals of confluent hypergeometric functions provides interesting subordination and superordination results and inspired the idea of using this operator to introduce a new class of analytic functions. Given the class of functions An=fHU:fz=z+an+1zn+1+,zU written simply A when n=1, the newly introduced class involves functions fA considered in the study due to their special properties. The aim of this paper is to present the outcomes of the study performed on the new class, which include a coefficient inequality, a distortion theorem and extreme points of the class. The starlikeness and convexity properties of this class are also discussed, and partial sums of functions from the class are evaluated in order to obtain class boundary properties. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
16 pages, 301 KiB  
Article
Radius of Star-Likeness for Certain Subclasses of Analytic Functions
by Caihuan Zhang, Mirajul Haq, Nazar Khan, Muhammad Arif, Khurshid Ahmad and Bilal Khan
Symmetry 2021, 13(12), 2448; https://doi.org/10.3390/sym13122448 - 19 Dec 2021
Cited by 2 | Viewed by 2559
Abstract
In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition fzgz>0, for some analytic function, g, with [...] Read more.
In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition fzgz>0, for some analytic function, g, with z+12nzgz>0,nN. We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
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