scholarly journals On the Connection Problem for Painlevé Differential Equation in View of Geometric Function Theory

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1198
Author(s):  
Rabha W. Ibrahim ◽  
Rafida M. Elobaid ◽  
Suzan J. Obaiys
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Asymptotic Analysis ◽  
Function Theory ◽  
Symmetric Domain ◽  
Geometric Function Theory ◽  
Limiting Behavior ◽  
Connection Problem ◽  
Geometric Function ◽  
Univalent Solution

Asymptotic analysis is a branch of mathematical analysis that describes the limiting behavior of the function. This behavior appears when we study the solution of differential equations analytically. The recent work deals with a special class of third type of Painlevé differential equation (PV). Our aim is to find asymptotic, symmetric univalent solution of this class in a symmetric domain with respect to the real axis. As a result that the most important problem in the asymptotic expansion is the connections bound (coefficients bound), we introduce a study of this problem.

scholarly journals Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials

2021 ◽  
pp. 1-26
Author(s):  
ELENA CHERKAEV ◽  
MINWOO KIM ◽  
MIKYOUNG LIM
Keyword(s):  
Composite Materials ◽  
Series Expansion ◽  
Function Theory ◽  
Effective Conductivity ◽  
Two Dimensions ◽  
Boundary Integral ◽  
Geometric Function Theory ◽  
Boundary Integral Formulation ◽  
Geometric Function ◽  
Series Expression

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

scholarly journals Integral transforms for logharmonic mappings

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hugo Arbeláez ◽  
Víctor Bravo ◽  
Rodrigo Hernández ◽  
Willy Sierra ◽  
Osvaldo Venegas
Keyword(s):  
Function Theory ◽  
Classical Problem ◽  
Integral Transforms ◽  
Geometric Function Theory ◽  
Integral Transformations ◽  
Geometric Function

AbstractBieberbach’s conjecture was very important in the development of geometric function theory, not only because of the result itself, but also due to the large amount of methods that have been developed in search of its proof. It is in this context that the integral transformations of the type $f_{\alpha }(z)=\int _{0}^{z}(f(\zeta )/\zeta )^{\alpha }\,d\zeta $ f α ( z ) = ∫ 0 z ( f ( ζ ) / ζ ) α d ζ or $F_{\alpha }(z)=\int _{0}^{z}(f'(\zeta ))^{\alpha }\,d\zeta $ F α ( z ) = ∫ 0 z ( f ′ ( ζ ) ) α d ζ appear. In this note we extend the classical problem of finding the values of $\alpha \in \mathbb{C}$ α ∈ C for which either $f_{\alpha }$ f α or $F_{\alpha }$ F α are univalent, whenever f belongs to some subclasses of univalent mappings in $\mathbb{D}$ D , to the case of logharmonic mappings by considering the extension of the shear construction introduced by Clunie and Sheil-Small in (Clunie and Sheil-Small in Ann. Acad. Sci. Fenn., Ser. A I 9:3–25, 1984) to this new scenario.

scholarly journals Some estimates for extremal decomposition of the complex plane

2018 ◽  
Vol 32 ◽  
pp. 42-47
Author(s):  
Iryna Denega
Keyword(s):  
Complex Variable ◽  
Open Problem ◽  
Function Theory ◽  
Extremal Problems ◽  
Geometric Function Theory ◽  
Geometric Function ◽  
Inner Radius ◽  
Overlapping Domains ◽  
Disjoint Domains

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. In this paper, we consider the well-known problem of maximum of the functional \(r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right)\), where \(B_{0}\),..., \(B_{n}\) are pairwise disjoint domains in \(\overline{\mathbb{C}}\), \( a_0=0 \), \(|a_{k}|=1\), \(k=\overline{1,n}\) are different points of the circle, \(\gamma\in (0, n]\), and \(r(B,a)\) is the inner radius of the domain \(B\subset\overline{\mathbb{C}}\) relative to the point \( a \). This problem was posed as an open problem in the Dubinin paper in 1994. Till now, this problem has not been solved, though some partial solutions are available. In the paper an estimate for the inner radius of the domain that contains the point zero is found. The main result of the paper generalizes the analogous results of [1, 2] to the case of an arbitrary arrangement of systems of points on \(\overline{\mathbb{C}}\).

scholarly journals New Trends in Geometric Function Theory 2011

2012 ◽  
Vol 2012 ◽  
pp. 1-2
Author(s):  
Teodor Bulboacă ◽  
Nak Eun Cho ◽  
Stanisława R. Kanas
Keyword(s):  
Function Theory ◽  
Geometric Function Theory ◽  
Geometric Function
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