Symmetry in Geometric Functions and Mathematical Analysis

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

Deadline for manuscript submissions: closed (30 April 2021) | Viewed by 22189

Special Issue Editors


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Guest Editor
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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Guest Editor
Tohoku University
Interests: complex analysis and applications

Special Issue Information

Dear Colleagues,

This Special Issue “Symmetry in Geometric Functions and Mathematical Analysis” is devoted to the publication of high-quality research, especially relating to its 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 theory of complex analysis of one and several variables, and application in algebraic geometry, number theory, as well as 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
Prof. Dr. Toshiyuki Sugawa
Guest Editors

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. Symmetry 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

  • 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 (12 papers)

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Research

27 pages, 503 KiB  
Article
Holomorphic Extensions Associated with Fourier–Legendre Series and the Inverse Scattering Problem
by Enrico De Micheli
Symmetry 2021, 13(6), 1009; https://doi.org/10.3390/sym13061009 - 4 Jun 2021
Viewed by 1256
Abstract
In this paper, we consider the inverse scattering problem and, in particular, the problem of reconstructing the spectral density associated with the Yukawian potentials from the sequence of the partial-waves f of the Fourier–Legendre expansion of the scattering amplitude. We prove that [...] Read more.
In this paper, we consider the inverse scattering problem and, in particular, the problem of reconstructing the spectral density associated with the Yukawian potentials from the sequence of the partial-waves f of the Fourier–Legendre expansion of the scattering amplitude. We prove that if the partial-waves f satisfy a suitable Hausdorff-type condition, then they can be uniquely interpolated by a function f˜(λ)C, analytic in a half-plane. Assuming also the Martin condition to hold, we can prove that the Fourier–Legendre expansion of the scattering amplitude converges uniformly to a function f(θ)C (θ being the complexified scattering angle), which is analytic in a strip contained in the θ-plane. This result is obtained mainly through geometrical methods by replacing the analysis on the complex cosθ-plane with the analysis on a suitable complex hyperboloid. The double analytic symmetry of the scattering amplitude is therefore made manifest by its analyticity properties in the λ- and θ-planes. The function f(θ) is shown to have a holomorphic extension to a cut-domain, and from the discontinuity across the cuts we can iteratively reconstruct the spectral density σ(μ) associated with the class of Yukawian potentials. A reconstruction algorithm which makes use of Pollaczeck and Laguerre polynomials is finally given. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)
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8 pages, 266 KiB  
Article
Dense and σ-Porous Subsets in Some Families of Darboux Functions
by Gertruda Ivanova and Irena Domnik
Symmetry 2021, 13(5), 759; https://doi.org/10.3390/sym13050759 - 27 Apr 2021
Cited by 3 | Viewed by 1126
Abstract
G. Ivanova and E. Wagner-Bojakowska shown that the set of Darboux quasi-continuous functions with nowhere dense set of discontinuity points is dense in the metric space of Darboux quasi-continuous functions with the supremum metric. We prove that this set also is σ-strongly [...] Read more.
G. Ivanova and E. Wagner-Bojakowska shown that the set of Darboux quasi-continuous functions with nowhere dense set of discontinuity points is dense in the metric space of Darboux quasi-continuous functions with the supremum metric. We prove that this set also is σ-strongly porous in such space. We obtain the symmetrical result for the family of strong Świątkowski functions, i.e., that the family of strong Świątkowski functions with nowhere dense set of discontinuity points is dense (thus, “large”) and σ-strongly porous (thus, asymmetrically, “small”) in the family of strong Świątkowski functions. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)
18 pages, 370 KiB  
Article
Applications of Certain Conic Domains to a Subclass of q-Starlike Functions Associated with the Janowski Functions
by Bilal Khan, Hari Mohan Srivastava, Nazar Khan, Maslina Darus, Qazi Zahoor Ahmad and Muhammad Tahir
Symmetry 2021, 13(4), 574; https://doi.org/10.3390/sym13040574 - 31 Mar 2021
Cited by 25 | Viewed by 2330
Abstract
In our present investigation, with the help of the basic (or q-) calculus, we first define a new domain which involves the Janowski function. We also define a new subclass of the class of q-starlike functions, which maps the open unit [...] Read more.
In our present investigation, with the help of the basic (or q-) calculus, we first define a new domain which involves the Janowski function. We also define a new subclass of the class of q-starlike functions, which maps the open unit disk U, given by U= z:zC and z <1, onto this generalized conic type domain. We study here some such potentially useful results as, for example, the sufficient conditions, closure results, the Fekete-Szegö type inequalities and distortion theorems. We also obtain the lower bounds for the ratio of some functions which belong to this newly-defined function class and for the sequences of the partial sums. Our results are shown to be connected with several earlier works related to the field of our present investigation. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward (p,q)-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter p is obviously redundant. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)
12 pages, 312 KiB  
Article
Second Hankel Determinant for a Certain Subclass of Bi-Close to Convex Functions Defined by Kaplan
by Stanislawa Kanas, Pesse V. Sivasankari, Roy Karthiyayini and Srikandan Sivasubramanian
Symmetry 2021, 13(4), 567; https://doi.org/10.3390/sym13040567 - 29 Mar 2021
Cited by 4 | Viewed by 1538
Abstract
In this paper, we consider the class of strongly bi-close-to-convex functions of order α and bi-close-to-convex functions of order β. We obtain an upper bound estimate for the second Hankel determinant for functions belonging to these classes. The results in this article [...] Read more.
In this paper, we consider the class of strongly bi-close-to-convex functions of order α and bi-close-to-convex functions of order β. We obtain an upper bound estimate for the second Hankel determinant for functions belonging to these classes. The results in this article improve some earlier result obtained for the class of bi-convex functions. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)
18 pages, 351 KiB  
Article
Growth Analysis of Meromorphic Solutions of Linear Difference Equations with Entire or Meromorphic Coefficients of Finite φ-Order
by Junesang Choi, Sanjib Kumar Datta and Nityagopal Biswas
Symmetry 2021, 13(2), 267; https://doi.org/10.3390/sym13020267 - 5 Feb 2021
Cited by 2 | Viewed by 1573
Abstract
Many researchers’ attentions have been attracted to various growth properties of meromorphic solution f (of finite φ-order) of the following higher order linear difference equation [...] Read more.
Many researchers’ attentions have been attracted to various growth properties of meromorphic solution f (of finite φ-order) of the following higher order linear difference equation Anzfz+n+...+A1zfz+1+A0zfz=0, where Anz,,A0z are entire or meromorphic coefficients (of finite φ-order) in the complex plane (φ:[0,)(0,) is a non-decreasing unbounded function). In this paper, by introducing a constant b (depending on φ) defined by lim̲rlogrlogφ(r)=b<, and we show how nicely diverse known results for the meromorphic solution f of finite φ-order of the above difference equation can be modified. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)
10 pages, 262 KiB  
Article
New Conditions for Univalence of Confluent Hypergeometric Function
by Georgia Irina Oros
Symmetry 2021, 13(1), 82; https://doi.org/10.3390/sym13010082 - 5 Jan 2021
Cited by 15 | Viewed by 2172
Abstract
Since in many particular cases checking directly the conditions from the definitions of starlikeness or convexity of a function can be difficult, in this paper we use the theory of differential subordination and in particular the method of admissible functions in order to [...] Read more.
Since in many particular cases checking directly the conditions from the definitions of starlikeness or convexity of a function can be difficult, in this paper we use the theory of differential subordination and in particular the method of admissible functions in order to determine conditions of starlikeness and convexity for the confluent (Kummer) hypergeometric function of the first kind. Having in mind the results obtained by Miller and Mocanu in 1990 who used a,cR, for the confluent (Kummer) hypergeometric function, in this investigation a and c complex numbers are used and two criteria for univalence of the investigated function are stated. An example is also included in order to show the relevance of the original results of the paper. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)
15 pages, 315 KiB  
Article
Estimates of Coefficient Functionals for Functions Convex in the Imaginary-Axis Direction
by Paweł Zaprawa and Katarzyna Tra̧bka-Wiȩcław
Symmetry 2020, 12(10), 1736; https://doi.org/10.3390/sym12101736 - 20 Oct 2020
Viewed by 1462
Abstract
Let C0(h) be a subclass of analytic and close-to-convex functions defined in the open unit disk by the formula Re{(1z2)f(z)}>0. In this [...] Read more.
Let C0(h) be a subclass of analytic and close-to-convex functions defined in the open unit disk by the formula Re{(1z2)f(z)}>0. In this paper, some coefficient problems for C0(h) are considered. Some properties and bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates of the difference and of sum of successive coefficients, bounds of the sum of the first n coefficients and bounds of the n-th coefficient. The obtained results are used to determine coefficient estimates for both functions convex in the imaginary-axis direction with real coefficients and typically real functions. Moreover, the sum of the first initial coefficients for functions with a positive real part and with a fixed second coefficient is estimated. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)
10 pages, 239 KiB  
Article
Some Results of Fekete-Szegö Type. Results for Some Holomorphic Functions of Several Complex Variables
by Renata Długosz and Piotr Liczberski
Symmetry 2020, 12(10), 1707; https://doi.org/10.3390/sym12101707 - 16 Oct 2020
Cited by 2 | Viewed by 1245
Abstract
This paper is devoted to a generalization of the well-known Fekete-Szegö type coefficients problem for holomorphic functions of a complex variable onto holomorphic functions of several variables. The considerations concern three families of such functions f, which are bounded, having positive real [...] Read more.
This paper is devoted to a generalization of the well-known Fekete-Szegö type coefficients problem for holomorphic functions of a complex variable onto holomorphic functions of several variables. The considerations concern three families of such functions f, which are bounded, having positive real part and which Temljakov transform Lf has positive real part, respectively. The main result arise some sharp estimates of the Minkowski balance of a combination of 2-homogeneous and the square of 1-homogeneous polynomials occurred in power series expansion of functions from aforementioned families. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)
6 pages, 241 KiB  
Article
Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case
by Jan L. Cieśliński and Dzianis Zhalukevich
Symmetry 2020, 12(9), 1550; https://doi.org/10.3390/sym12091550 - 20 Sep 2020
Cited by 5 | Viewed by 1869
Abstract
Scators form a vector space endowed with a non-distributive product, in the hyperbolic case, have physical applications related to some deformations of special relativity (breaking the Lorentz symmetry) while the elliptic case leads to new examples of hypercomplex numbers and related notions of [...] Read more.
Scators form a vector space endowed with a non-distributive product, in the hyperbolic case, have physical applications related to some deformations of special relativity (breaking the Lorentz symmetry) while the elliptic case leads to new examples of hypercomplex numbers and related notions of holomorphicity. Until now, only a few particular cases of scator holomorphic functions have been found. In this paper we obtain all solutions of the generalized Cauchy–Riemann system which describes analogues of holomorphic functions in the (1+2)-dimensional scator space. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)
7 pages, 712 KiB  
Article
New Identities Dealing with Gauss Sums
by Wenpeng Zhang, Abdul Samad and Zhuoyu Chen
Symmetry 2020, 12(9), 1416; https://doi.org/10.3390/sym12091416 - 26 Aug 2020
Cited by 6 | Viewed by 1698
Abstract
In this article, we used the elementary methods and the properties of the classical Gauss sums to study the problem of calculating some Gauss sums. In particular, we obtain some interesting calculating formulas for the Gauss sums corresponding to the eight-order and twelve-order [...] Read more.
In this article, we used the elementary methods and the properties of the classical Gauss sums to study the problem of calculating some Gauss sums. In particular, we obtain some interesting calculating formulas for the Gauss sums corresponding to the eight-order and twelve-order characters modulo p, where p be an odd prime with p=8k+1 or p=12k+1. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)
8 pages, 275 KiB  
Article
The Fifth Coefficient of Bazilevič Functions
by Oh Sang Kwon, Derek Keith Thomas and Young Jae Sim
Symmetry 2020, 12(8), 1228; https://doi.org/10.3390/sym12081228 - 26 Jul 2020
Viewed by 1468
Abstract
Let fA, the class of normalized analytic functions defined in the unit disk D, and be given by f(z)=z+n=2anzn for zD. [...] Read more.
Let fA, the class of normalized analytic functions defined in the unit disk D, and be given by f(z)=z+n=2anzn for zD. This paper presents a new approach to finding bounds for |an|. As an application, we find the sharp bound for |a5| for the class B1(α) of Bazilevič functions when α1. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)
11 pages, 2936 KiB  
Article
Subclasses of Starlike and Convex Functions Associated with the Limaçon Domain
by Vali Soltani Masih and Stanisława Kanas
Symmetry 2020, 12(6), 942; https://doi.org/10.3390/sym12060942 - 3 Jun 2020
Cited by 34 | Viewed by 3320
Abstract
Let ST L ( s ) and CV L ( s ) denote the family of analytic and normalized functions f in the unit disk D : = z : | z | < 1 , such that the quantity [...] Read more.
Let ST L ( s ) and CV L ( s ) denote the family of analytic and normalized functions f in the unit disk D : = z : | z | < 1 , such that the quantity z f ( z ) / f ( z ) or 1 + z f ( z ) / f ( z ) respectively are lying in the region bounded by the limaçon ( u 1 ) 2 + v 2 s 4 2 = 4 s 2 u 1 + s 2 2 + v 2 , where 0 < s 1 / 2 . The limaçon of Pascal is a curve that possesses properties which qualify it for the several applications in mathematics, statistics (hypothesis testing problem) but also in mechanics (fluid processing applications, known limaçon technology is employed to extract electrical power from low-grade heat, etc.). In this paper we present some results concerning the behavior of f on the classes ST L ( s ) or CV L ( s ) . Some appropriate examples are given. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis)
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