Geometric Function Theory Research Articles

Connecting quantum calculus and harmonic starlike functions

Quantum calculus or q-calculus plays an important role in hypergeometric series, quantum physics, operator theory, approximation theory, sobolev spaces, geometric functions theory and others. But role of q-calculus in the theory of harmonic univalent functions is quite new. In this paper, we make an attempt to connect quantum calculus and harmonic univalent starlike functions. In particular, we introduce and investigate the properties of q-harmonic functions and q-harmonic starlike functions of order ?.

Inclusion and convolution features of univalent meromorphic functions correlating with Mittag-Leffler function

The so-called Mittag-Leffler function (M-LF) provides solutions to the fractional differential or integral equations with numerous implementations in applied sciences and other allied disciplines. During the previous century, the interest in M-LF has significantly developed and a variety of extensions and generalizations forms of the M-LF have been posed. Moreover, M-LF played a distinguished and important role in Geometric Function Theory (GFT). The intent of the current study is to reveal various inclusion and convolution features for a specific subclass of univalent meromorphic functions correlating with the integrodifferential operator containing an extended generalized M-LF. Some consequences of the major geometric outcomes are also presented.

Coefficient estimates of certain subclasses of analytic functions associated with Hohlov operator

A typical quandary in geometric functions theory is to study a functional composed of amalgamations of the coefficients of the pristine function. Conventionally, there is a parameter over which the extremal value of the functional is needed. The present paper deals with consequential functional of this type. By making use of Hohlov operator, a new subclass [Formula: see text] of analytic functions defined ins the open unit disk is introduced. For both real and complex parameter, the sharp bounds for the Fekete–Szegö problems are found. An attempt has also been taken to found the sharp upper bound to the second and third Hankel determinant for functions belonging to this class. All the extremal functions are express in term of Gauss hypergeometric function and convolution. Finally, the sufficient condition for functions to be in [Formula: see text] is derived. Relevant connections of the new results with well-known ones are pointed out.

On mappings with finite length distortion on Riemann surfaces

The present paper is a natural continuation of our previous paper (2017) on the boundary behavior of mappings in the Sobolev classes on Riemann surfaces, where the reader will be able to find the corresponding historic comments and a discussion of many definitions and relevant results. The given paper was devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec on Riemannian surfaces first introduced for the plane in the paper of Iwaniec T. and Sverak V. (1993) On mappings with integrable dilatation and then extended to the spatial case in the monograph of Iwaniec T. and Martin G. (2001) devoted to Geometric function theory and non-linear analysis. At the present paper, it is developed the theory of the boundary behavior of the so--called mappings with finite length distortion first introduced in the paper of Martio O., Ryazanov V., Srebro U. and Yakubov~E. (2004) in the spatial case, see also Chapter 8 in their monograph (2009) on Moduli in modern mapping theory. As it was shown in the paper of Kovtonyuk D., Petkov I. and Ryazanov V. (2017) On the boundary behavior of mappings with finite distortion in the plane, such mappings, generally speaking, are not mappings with finite distortion by Iwaniec because their first partial derivatives can be not locally integrable. At the same time, this class is a generalization of the known class of mappings with bounded distortion by Martio--Vaisala from their paper (1988). Moreover, this class contains as a subclass the so-called finitely bi-Lipschitz mappings introduced for the spatial case in the paper of Kovtonyuk D. and Ryazanov V. (2011) On the boundary behavior of generalized quasi-isometries, that in turn are a natural generalization of the well-known classes of bi-Lipschitz mappings as well as isometries and quasi-isometries. In the research of the local and boundary behavior of mappings with finite length distortion in the spatial case, the key fact was that they satisfy some modulus inequalities which was a motivation for the consideration more wide classes of mappings, in particular, the Q-homeomorphisms (2005) and the mappings with finite area distortion (2008). Hence it is natural that under the research of mappings with finite length distortion on Riemann surfaces we start from establishing the corresponding modulus inequalities that are the main tool for us. On this basis, we prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extension to the boundary of the mappings with finite length distortion between domains on arbitrary Riemann surfaces.

Estimation of the products of the inner radii of domains with an additional symmetry condition

In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. Based on these elementary estimates a number of new estimates for functions realizing a conformal mapping of a disc onto domains with certain special properties are obtained. Estimates of this type are fundamental to solving some metric problems arising when considering the cor\-res\-pon\-dence of boundaries under a conformal mapping. Also, on the basis of the results concerning various extremal properties of conformal mappings, the problem of the representability of functions of a complex variable by a uniformly convergent series of polynomials is solved. In this paper, we consider the problem on maximum the products of the inner radii of $n$ disjoint domains with an additional symmetry condition that contain points of extended complex plane and the degree $\gamma$ of the inner radius of the domain that contains the zero point. An upper estimate for the maximum of this product is found for all values of $\gamma\in(0,\,n]$. The main result of the paper generalizes and strengthens the results of the predecessors [1-4] for the case of an arbitrary arrangement of points systems on $\overline{\mathbb{C}}$. In proving the main theorem, the arguments of proving of Lemma 1 [5] and the ideas of proving Theorem 1 [3] played a key role. We also established the conditions under which the structure of points and domains is not important. The corresponding results are obtained for the case when the points are placed on the unit circle and in the case of any fixed $n$-radial system of points.

(p, q)-Lucas polynomials and their applications to bi-univalent functions

In the present paper, by using the Lp,q,n(x) functions, our methodology intertwine to yield the Theory of Geometric Functions and that of Special Functions, which are usually considered as very different fields. Thus, we aim at introducing a new class of bi-univalent functions defined through the (p, q)-Lucas polynomials. Furthermore, we derive coefficient inequalities and obtain Fekete-Szego problem for this new function class.

About the Cover: The Ruscheweyh Derivatives

Honoring Stephan Ruscheweyh’s magnificent contributions to complex analysis in general and to geometric function theory in particular, the cover of this volume shows a (modified) phase plot of a Ruscheweyh derivative of the complex tangent function. In this note we sketch some background.

Simple criteria for univalence and coefficient bounds for a certain subclass of analytic functions

In the first part of this work we present several new geometric properties of analytic functions by applying the differential subordination. In addition, several results in the geometric functions theory pointed out. In the second part we find upper bounds for coefficients of functions in class $\mathcal{B}_\Sigma^{q,\mu}(\beta,\lambda,h)$ which is defined by fractional $q$-calculus operators.

On certain subclasses of uniformly convex functions

Geometric Function Theory is a remarkable area in complex analysis and it plays a significant role and contributes new ideas and results even today. The study of univalent and multivalent functions is one of the main branches of geometric function theory and one of the most fascinating aspects of the theory of analytical functions of a complex variable.The purpose of this paper is to introduce new classes of uniformly convex functions Rp(γ, α, β, q) and KRp(γ, α, β, q) in U. Also obtain some sufficient and necessary conditions for the function belongs to these classes. As well as, we investigate some geometri

Geometric properties of the nonlinear resolvent of holomorphic generators

Let f be the infinitesimal generator of a one-parameter semigroup of holomorphic self-mappings of the open unit disk Δ. Our main purpose is to study properties of the family R of non-linear resolvents (I+rf)−1:Δ→Δ,r≥0, in the spirit of classical geometric function theory. To make a connection with this theory, we mostly consider the case where f(0)=0 and f′(0) is a positive real number. We found, in particular, that R forms an inverse Löwner chain of hyperbolically convex functions. Moreover, each element of R satisfies the Noshiro-Warschawski condition. This, in turn, implies that each element of R is also the infinitesimal generator of a one-parameter semigroup on Δ. We consider also quasiconformal extensions of elements of R. Finally we study the existence of repelling fixed points of this family.

Estimates of the inner radii of non-overlapping domains

Some extremal problems of the geometric theory of functions of a complex variable related to the estimates of functionals defined on systems of non-overlapping domains are considered. Till now, many such problems have not been solved, though some partial solutions are available. In the paper, the improved method is proposed for solving the problems on extremal decomposition of the complex plane. The main results generalize and strengthen some known results in the theory of non-overlapping domains with free poles to the case of an arbitrary arrangement of systems of points on the complex plane.

Vortex motion and geometric function theory: the role of connections

We formulate the equations for point vortex dynamics on a closed two-dimensional Riemannian manifold in the language of affine and other kinds of connections. This can be viewed as a relaxation of standard approaches, using the Riemannian metric directly, to an approach based more on local coordinates provided with a minimal amount of extra structure. The speed of a vortex is then expressed in terms of the difference between an affine connection derived from the coordinate Robin function and the Levi–Civita connection associated with the Riemannian metric. A Hamiltonian formulation of the same dynamics is also given. The relevant Hamiltonian function consists of two main terms. One of the terms is the well-known quadratic form based on a matrix whose entries are Green and Robin functions, while the other term describes the energy contribution from those circulating flows which are not implicit in the Green functions. One main issue of the paper is a detailed analysis of the somewhat intricate exchanges of energy between these two terms of the Hamiltonian. This analysis confirms the mentioned dynamical equations formulated in terms of connections.This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

(P,Q)-Lucas polynomial coefficient inequalities of the bi-univalent function class

Recently, Lucas polynomials and other special polynomials gained importance in the field of geometric function theory. In this study, by connecting these polynomials, subordination, and the Al-Oboudi differential operator, we introduce a new class of bi-univalent functions and obtain coefficient estimates and Fekete-SzegÖ inequalities for this new class.

New Symmetric Differential and Integral Operators Defined in the Complex Domain

The symmetric differential operator is a generalization operating of the well-known ordinary derivative. These operators have advantages in boundary value problems, statistical studies and spectral theory. In this effort, we introduce a new symmetric differential operator (SDO) and its integral in the open unit disk. This operator is a generalization of the Sàlàgean differential operator. Our study is based on geometric function theory and its applications in the open unit disk. We formulate new classes of analytic functions using SDO depending on the symmetry properties. Moreover, we define a linear combination operator containing SDO and the Ruscheweyh derivative. We illustrate some inclusion properties and other inequalities involving SDO and its integral.

Extremal decomposition of a multidimensional complex space for five domains

The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin’s paper in the journal “Russian Mathematical Surveys” in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1; 2:57] and generalized this result to the case of multidimensional complex space.

Complex cellular structures

We introduce the notion of a complex cell, a complexification of the cells/cylinders used in real tame geometry. For $\delta\in(0,1)$ and a complex cell $\mathcal{C}$ we define its holomorphic extension $\mathcal{C}\subset\mathcal{C}^\delta$, which is again a complex cell. The hyperbolic geometry of $\mathcal{C}$ within $\mathcal{C}^\delta$ provides the class of complex cells with a rich geometric function theory absent in the real case. We use this to prove a complex analog of the cellular decomposition theorem of real tame geometry. In the algebraic case we show that the complexity of such decompositions depends polynomially on the degrees of the equations involved. Using this theory, we refine the Yomdin-Gromov algebraic lemma on $C^r$-smooth parametrizations of semialgebraic sets: we show that the number of $C^r$ charts can be taken to be polynomial in the smoothness order $r$ and in the complexity of the set. The algebraic lemma was initially invented in the work of Yomdin and Gromov to produce estimates for the topological entropy of $C^\infty$ maps. For analytic maps our refined version, combined with work of Burguet, Liao and Yang, establishes an optimal refinement of these estimates in the form of tight bounds on the tail entropy and volume growth. This settles a conjecture of Yomdin who proved the same result in dimension two in 1991. A self-contained proof of these estimates using the refined algebraic lemma is given in an appendix by Yomdin. The algebraic lemma has more recently been used in the study of rational points on algebraic and transcendental varieties. We use the theory of complex cells in these two directions. In the algebraic context we refine a result of Heath-Brown on interpolating rational points in algebraic varieties. In the transcendental context we prove an interpolation result for (unrestricted) logarithmic images of subanalytic sets.

Application of Dzyadyk’s Polynomial Kernels in the Constructive Function Theory

This is a survey of recent results in the constructive theory of functions of complex variable obtained by the author through the application of the theory of Dzyadyk’s kernels combined with the methods and results from modern geometric function theory and the theory of quasiconformal mappings.

14th International Symposium on Geometric Function Theory and Applications

14th International Symposium on Geometric Function Theory and Applications (GFTA 2018)3-5 December 2018, Puri Pujangga Hotel, Universiti Kebangsaan Malaysia, Selangor, MalaysiaThe 14th International Symposium on Geometric Function Theory and Applications (GFTA 2018) is a continuation of the series of the annual international “GFTA” symposia. GFTA 2018 is the 14th symposium in its series. The first symposium was held in 2005 at the University of Victoria, Canada. Since then, the symposium was held annually at different locations, mainly in Romania, Turkey, Bulgaria, Macedonia and Malaysia. The list of all the symposia can be found in the conference website. The scientific program will focus on Geometric Function Theory and its Applications, as well as other fields of Mathematical Sciences, particularly Pure and Applied Mathematics. Therefore, the main topics of the conference is divided into two main categories: Pure Mathematics and Applied Mathematics.The aim of the symposium is to bring together leading experts as well as young researchers working on topics mainly related to Univalent and Geometric Function Theory along with other areas of Pure and Applied Mathematics and to present their recent works to the mathematical sciences community. Notable speakers in the field of Geometric Function Theory and other fields of mathematical sciences are invited to give their plenary talks. The Symposium is also open to other fields. Thus, it is with great pleasure for the School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (the National University of Malaysia) together with the Malaysian Mathematical Sciences Society (PERSAMA) to organise the 14th International Symposium on Geometric Function Theory and Applications (GFTA 2018) to bridge as well as nurture the understanding and collaboration amongst the regional and global mathematical scientists and practitioners.EditorsRoslinda NazarMaslina DarusAlawiah IbrahimList of Organising Committee, Keynote Speakers and Invited Speakers (Parallel Sessions) are available in this pdf.

Fekete-Szegö Problem of Functions Associated with Hyperbolic Domains

In the field of Geometric Function Theory, one can not deny the importance of analytic and univalent functions. The characteristics of these functions including their taylor series expansion, their coefficients in these representations as well as their associated functional inequalities have always attracted the researchers. In particular, Fekete-Szego inequality is one of such vastly studied and investigated functional inequality. Our main focus in this article is to investigate the Fekete-Szego functional for the class of analytic functions associated with hyperbolic regions. Tofurther enhance the worth of our work, we include similar problems for the inverse functions of these discussed analytic functions.

Inclusion properties of Lucas polynomials for bi-univalent functions introduced through the q-analogue of the Noor integral operator

In this paper, by using the $(\mathbf{P},\mathbf{Q})$-Lucas polynomials and the $\mathfrak{q}$-analogue of the Noor integral operator, we aim to build a bridge between the theory of geometric functions and that of special functions.