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Special Issue "Complex Analysis"

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

Deadline for manuscript submissions: closed (30 November 2021) | Viewed by 12017

Special Issue Editor

Prof. Dr. Andriy Bandura
E-Mail Website1 Website2
Guest Editor
Department of Advanced Mathematics, Ivano-Frankivsk National Technical University of Oil and Gas, 76019 Ivano-Frankivsk, Ukraine
Interests: entire function; analytic function; growth estimates; bounded index; bounded index in direction; bounded index in joint variables; slice holomorphic function; unit ball; polydisc; vector-valued analytic function; unit disc; value distribution; analytic solution; complex differential equation
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We are excited to launch a new Special Issue of Axioms. The central topic in the Special Issue will be “Complex Analysis and Its Applications”. Our aim is to showcase recent contributions in the many branches of both theoretical and practical studies in complex analysis, as well as its extensions and generalizations. Modern complex analysis is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics, approximation theory, ordinary and partial differential equations, and their systems. Complex analysis also has many applications in engineering fields and physics.

Among the topics that this Special Issue will address, we may consider the following non-exhaustive list: analytic functions; applications of complex analysis; analytic theory of differential equations; geometric functions theory; Dirichlet series; and meromorphic functions. The Special Issue is open to receiving further related topics of complex analysis.

We highly encourage you to submit your current original research papers for inclusion in the Special Issue.

Prof. Dr. Andriy Bandura
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 1600 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

  • Entire functions of several variables
  • Analytic functions of several variables
  • Growth estimates
  • Applications of complex analysis
  • Unit ball
  • Polydisc
  • Geometric function theory
  • Analytic solutions of differential equations
  • Meromorphic functions
  • Nevanlinna theory
  • Dirichlet series
  • Approximation theory in the complex plane
  • Entire curves
  • Vector-valued entire functions

Published Papers (11 papers)

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Research

Article
Vector-Valued Entire Functions of Several Variables: Some Local Properties
Axioms 2022, 11(1), 31; https://doi.org/10.3390/axioms11010031 - 15 Jan 2022
Cited by 1 | Viewed by 1117
Abstract
The present paper is devoted to the properties of entire vector-valued functions of bounded L-index in join variables, where L:CnR+n is a positive continuous function. For vector-valued functions from this class we prove some propositions [...] Read more.
The present paper is devoted to the properties of entire vector-valued functions of bounded L-index in join variables, where L:CnR+n is a positive continuous function. For vector-valued functions from this class we prove some propositions describing their local properties. In particular, these functions possess the property that maximum of norm for some partial derivative at a skeleton of polydisc does not exceed norm of the derivative at the center of polydisc multiplied by some constant. The converse proposition is also true if the described inequality is satisfied for derivative in each variable. Full article
(This article belongs to the Special Issue Complex Analysis)
Article
On Complex Numbers in Higher Dimensions
Axioms 2022, 11(1), 22; https://doi.org/10.3390/axioms11010022 - 07 Jan 2022
Cited by 1 | Viewed by 617
Abstract
The geometric approach to generalized complex and three-dimensional hyper-complex numbers and more general algebraic structures being based upon a general vector space structure and a geometric multiplication rule which was only recently developed is continued here in dimension four and above. To this [...] Read more.
The geometric approach to generalized complex and three-dimensional hyper-complex numbers and more general algebraic structures being based upon a general vector space structure and a geometric multiplication rule which was only recently developed is continued here in dimension four and above. To this end, the notions of geometric vector product and geometric exponential function are extended to arbitrary finite dimensions and some usual algebraic rules known from usual complex numbers are replaced with new ones. An application for the construction of directional probability distributions is presented. Full article
(This article belongs to the Special Issue Complex Analysis)
Article
Wiman’s Type Inequality in Multiple-Circular Domain
Axioms 2021, 10(4), 348; https://doi.org/10.3390/axioms10040348 - 17 Dec 2021
Cited by 2 | Viewed by 1037
Abstract
In the paper we prove for the first time an analogue of the Wiman inequality in the class of analytic functions fA0p(G) in an arbitrary complete Reinhard domain GCp, pN [...] Read more.
In the paper we prove for the first time an analogue of the Wiman inequality in the class of analytic functions fA0p(G) in an arbitrary complete Reinhard domain GCp, pN represented by the power series of the form f(z)=f(z1,,zp)=n=0+anzn with the domain of convergence G. We have proven the following statement: If fAp(G) and hHp, then for a given ε=(ε1,,εp)R+p and arbitrary δ>0 there exists a set E|G| such that EΔεh(r)dr1drpr1rp<+ and for all rΔεE we have Mf(r)μf(r)(h(r))p+12lnp2+δh(r)lnp2+δ{μf(r)h(r)}j=1p(lnerjεj)p12+δ. Note, that this assertion at p=1,G=C,h(r)const implies the classical Wiman–Valiron theorem for entire functions and at p=1, the G=D:={zC:|z|<1},h(r)1/(1r) theorem about the Kővari-type inequality for analytic functions in the unit disc D; p>1 implies some Wiman’s type inequalities for analytic functions of several variables in Cn×Dk, n,kZ+,n+kN. Full article
(This article belongs to the Special Issue Complex Analysis)
Article
Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals
Axioms 2021, 10(4), 340; https://doi.org/10.3390/axioms10040340 - 10 Dec 2021
Cited by 2 | Viewed by 1568
Abstract
We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal [...] Read more.
We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities. Full article
(This article belongs to the Special Issue Complex Analysis)
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Article
On Multivalent Analytic Functions Considered by a Multi-Arbitrary Differential Operator in a Complex Domain
Axioms 2021, 10(4), 315; https://doi.org/10.3390/axioms10040315 - 23 Nov 2021
Viewed by 625
Abstract
(1) Background: There is an increasing amount of information in complex domains, which necessitates the development of various kinds of operators, such as differential, integral, and linear convolution operators. Few investigations of the fractional differential and integral operators of a complex variable have [...] Read more.
(1) Background: There is an increasing amount of information in complex domains, which necessitates the development of various kinds of operators, such as differential, integral, and linear convolution operators. Few investigations of the fractional differential and integral operators of a complex variable have been undertaken. (2) Methods: In this effort, we aim to present a generalization of a class of analytic functions based on a complex fractional differential operator. This class is defined by utilizing the subordination and superordination theory. (3) Results: We illustrate different fractional inequalities of starlike and convex formulas. Moreover, we discuss the main conditions to obtain sandwich inequalities involving the fractional operator. (4) Conclusion: We indicate that the suggested class is a generalization of recent works and can be applied to discuss the upper and lower bounds of a special case of fractional differential equations. Full article
(This article belongs to the Special Issue Complex Analysis)
Article
Certain Subclasses of Analytic Multivalent Functions Associated with Petal-Shape Domain
Axioms 2021, 10(4), 291; https://doi.org/10.3390/axioms10040291 - 03 Nov 2021
Cited by 7 | Viewed by 814
Abstract
In this article, we introduce a new class of multivalent analytic functions associated with petal-shape region. Furthermore, some useful properties, such as the Fekete–Szegö inequality, and their consequences for some special cases are discussed. For some specific value of function f, we [...] Read more.
In this article, we introduce a new class of multivalent analytic functions associated with petal-shape region. Furthermore, some useful properties, such as the Fekete–Szegö inequality, and their consequences for some special cases are discussed. For some specific value of function f, we obtain sufficient conditions for multivalent starlike functions connected with petal-shape domain. Finally, in the concluding section, we draw the attention of the interested readers toward the prospect of studying the basic or quantum (or q-) generalizations of the results, which are presented in this paper. However, the (p,q)-variations of the suggested q-results will provide a relatively minor and inconsequential development because the additional (rather forced-in) parameter p is obviously redundant. Full article
(This article belongs to the Special Issue Complex Analysis)
Article
Properties of Certain Multivalent Analytic Functions Associated with the Lemniscate of Bernoulli
Axioms 2021, 10(3), 160; https://doi.org/10.3390/axioms10030160 - 26 Jul 2021
Cited by 4 | Viewed by 897
Abstract
Using differential subordination, we consider conditions of β so that some multivalent analytic functions are subordinate to (1+z)γ (0<γ1). Notably, these results are applied to derive sufficient conditions for fA [...] Read more.
Using differential subordination, we consider conditions of β so that some multivalent analytic functions are subordinate to (1+z)γ (0<γ1). Notably, these results are applied to derive sufficient conditions for fA to satisfy the condition zf(z)f(z)21<1. Several previous results are extended. Full article
(This article belongs to the Special Issue Complex Analysis)
Article
Symmetry Breaking of a Time-2D Space Fractional Wave Equation in a Complex Domain
Axioms 2021, 10(3), 141; https://doi.org/10.3390/axioms10030141 - 30 Jun 2021
Cited by 2 | Viewed by 853
Abstract
(1) Background: symmetry breaking (self-organized transformation of symmetric stats) is a global phenomenon that arises in an extensive diversity of essentially symmetric physical structures. We investigate the symmetry breaking of time-2D space fractional wave equation in a complex domain; (2) Methods: a fractional [...] Read more.
(1) Background: symmetry breaking (self-organized transformation of symmetric stats) is a global phenomenon that arises in an extensive diversity of essentially symmetric physical structures. We investigate the symmetry breaking of time-2D space fractional wave equation in a complex domain; (2) Methods: a fractional differential operator is used together with a symmetric operator to define a new fractional symmetric operator. Then by applying the new operator, we formulate a generalized time-2D space fractional wave equation. We shall utilize the two concepts: subordination and majorization to present our results; (3) Results: we obtain different formulas of analytic solutions using the geometric analysis. The solution suggests univalent (1-1) in the open unit disk. Moreover, under certain conditions, it was starlike and dominated by a chaotic function type sine. In addition, the authors formulated a fractional time wave equation by using the Atangana–Baleanu fractional operators in terms of the Riemann–Liouville and Caputo derivatives. Full article
(This article belongs to the Special Issue Complex Analysis)
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Article
Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2
Axioms 2021, 10(2), 126; https://doi.org/10.3390/axioms10020126 - 21 Jun 2021
Viewed by 1116
Abstract
This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations [...] Read more.
This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), and the partial differential difference equations f(z+c)2+2αf(z+c)f(z)z1+f(z)z12=eg(z),f(z+c)2+2αf(z+c)f(z)z1+f(z)z2+f(z)z1+f(z)z22=eg(z). We establish some theorems about the forms of the finite order transcendental entire solutions of these functional equations. We also list a series of examples to explain the existence of the finite order transcendental entire solutions of such equations. Meantime, some examples show that there exists a very significant difference with the previous literature on the growth order of the finite order transcendental entire solutions. Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order. These results make a few improvements of the previous theorems given by Xu and Cao, Liu and Yang. Full article
(This article belongs to the Special Issue Complex Analysis)
Article
Properties of λ-Pseudo-Starlike Functions of Complex Order Defined by Subordination
Axioms 2021, 10(2), 86; https://doi.org/10.3390/axioms10020086 - 07 May 2021
Cited by 8 | Viewed by 950
Abstract
In this paper, we defined a new class of λ-pseudo-Bazilevič functions of complex order using subordination. Various classes of analytic functions that map unit discs onto a conic domain and some classes of special functions were studied in dual. Some subordination results, [...] Read more.
In this paper, we defined a new class of λ-pseudo-Bazilevič functions of complex order using subordination. Various classes of analytic functions that map unit discs onto a conic domain and some classes of special functions were studied in dual. Some subordination results, inequalities for the initial Taylor–Maclaurin coefficients and the unified solution of the Fekete–Szegő problem for subclasses of analytic functions related to various conic regions, are our main results. Our main results have many applications which are presented in the form of corollaries. Full article
(This article belongs to the Special Issue Complex Analysis)
Article
Relative Growth of Series in Systems of Functions and Laplace—Stieltjes-Type Integrals
Axioms 2021, 10(2), 43; https://doi.org/10.3390/axioms10020043 - 25 Mar 2021
Cited by 4 | Viewed by 818
Abstract
For a regularly converging-in-C series A(z)=n=1anf(λnz), where f is an entire transcendental function, the asymptotic behavior of the function [...] Read more.
For a regularly converging-in-C series A(z)=n=1anf(λnz), where f is an entire transcendental function, the asymptotic behavior of the function Mf1(MA(r)), where Mf(r)=max{|f(z)|:|z|=r}, is investigated. It is proven that, under certain conditions on the functions f, α, and the coefficients an, the equality limr+α(Mf1(MA(r)))α(r)=1 is correct. A similar result is obtained for the Laplace–Stiltjes-type integral I(r)=0a(x)f(rx)dF(x). Unresolved problems are formulated. Full article
(This article belongs to the Special Issue Complex Analysis)
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