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.

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Boundedness of fractional differential operator in complex spaces

Asian-European Journal of Mathematics ◽

10.1142/s1793557117500759 ◽

2017 ◽

Vol 10 (04) ◽

pp. 1750075 ◽

Cited By ~ 1

Author(s):

Rabha W. Ibrahim ◽

Adem Kılıçman ◽

Zainab E. Abdulnaby

Keyword(s):

Differential Operator ◽

Function Theory ◽

Geometric Function Theory ◽

Complex Spaces ◽

Geometric Function ◽

Fractional Differential Operator ◽

Fractional Differential

In this work, we introduce some properties of a complex fractional differential operator. The boundedness and some other properties are studied in view of geometric function theory.

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Differential Subordination Results for Certain Integrodifferential Operator and Its Applications

Abstract and Applied Analysis ◽

10.1155/2012/638234 ◽

2012 ◽

Vol 2012 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

M. A. Kutbi ◽

A. A. Attiya

Keyword(s):

Number Theory ◽

Zeta Function ◽

Function Theory ◽

Analytic Number Theory ◽

Differential Subordination ◽

Geometric Function Theory ◽

Integrodifferential Operator ◽

Analytic Number ◽

Geometric Function ◽

Lerch Zeta Function

We introduce an integrodifferential operatorJs,b(f) which plays an important role in theGeometric Function Theory. Some theorems in differential subordination forJs,b(f) are used. Applications inAnalytic Number Theoryare also obtained which give new results for Hurwitz-Lerch Zeta function and Polylogarithmic function.

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

European Journal of Applied Mathematics ◽

10.1017/s0956792521000127 ◽

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.

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Integral transforms for logharmonic mappings

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02578-y ◽

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.

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Some estimates for extremal decomposition of the complex plane

Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine ◽

10.37069/1683-4720-2018-32-5 ◽

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 $\overline{\mathbb{C}}$.

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