scholarly journals On starlike functions associated with cardioid domain

2021 ◽  
Vol 109 (123) ◽  
pp. 95-107
Author(s):  
Saira Zainab ◽  
Mohsan Raza ◽  
Janusz Sokół ◽  
Sarfraz Malik
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Function Theory ◽  
Starlike Functions ◽  
Geometric Function Theory ◽  
Geometrical Structures ◽  
Image Domain ◽  
Geometric Function ◽  
Comprehensive Study

Analytic functions are characterized by the geometry of their image domains. That?s why, geometry of image domain is of substantial importance to have a comprehensive study of analytic functions. To introduce and study new geometrical structures as image domain and to define their subsequent analytic functions is an ongoing part of research in geometric function theory. We introduced a new domain named as cardioid domain and defined the corresponding analytic function, see [14]. Here we further study the cardioid domain, to define and study starlike functions associated with cardioid domain.

scholarly journals Subordination for multivalent analytic functions associated with Wright generalized hypergeometric function.

2013 ◽  
Vol 44 (1) ◽  
pp. 61-71
Author(s):  
J. Sokol ◽  
N. Sarkar ◽  
P. Goswami ◽  
J. Dziok
Keyword(s):  
Hypergeometric Function ◽  
Analytic Functions ◽  
Function Theory ◽  
Geometric Function Theory ◽  
Generalized Hypergeometric Function ◽  
Geometric Function ◽  
Differential Subordinations

Recently M. K. Aouf and T. M. Seoudy, (2011, {\it Integral Trans. Spec. Func.} {\bf 22}(6) (2011), 423--430) have introduced families of analytic functions associated with the Dziok--Srivastava operator. In this work we use the Dziok--Raina operator to consider classes of multivalent analytic functions. It is connected with Wright generalized hypergeometric function, see J. Dziok and R. K. Raina (2004, {\it Demonstratio Math.}, {\bf 37}(3) 533--542). Moreover, we present a new result which extends some of the earlier results and give other properties of these classes. We have made use of differential subordinations and properties of convolution in geometric function theory.

Some properties associated to a certain class of starlike functions

2019 ◽  
Vol 69 (6) ◽  
pp. 1329-1340 ◽  
Author(s):  
Vali Soltani Masih ◽  
Ali Ebadian ◽  
Sibel Yalçin
Keyword(s):  
Function Theory ◽  
Starlike Functions ◽  
Sharp Inequality ◽  
Open Unit ◽  
Geometric Function Theory ◽  
Open Unit Disk ◽  
Geometric Function ◽  
The Family ◽  
Logarithmic Coefficients ◽  
Dilogarithm Function

Abstract Let 𝓐 denote the family of analytic functions f with f(0) = f′(0) – 1 = 0, in the open unit disk Δ. We consider a class $$\begin{array}{} \displaystyle \mathcal{S}^{\ast}_{cs}(\alpha):=\left\{f\in\mathcal{A} : \left(\frac{zf'(z)}{f(z)}-1\right)\prec \frac{z}{1+\left(\alpha-1\right) z-\alpha z^2}, \,\, z\in \Delta\right\}, \end{array}$$ where 0 ≤ α ≤ 1/2, and ≺ is the subordination relation. The methods and techniques of geometric function theory are used to get characteristics of the functions in this class. Further, the sharp inequality for the logarithmic coefficients γn of f ∈ $\begin{array}{} \mathcal{S}^{\ast}_{cs} \end{array}$(α): $$\begin{array}{} \displaystyle \sum_{n=1}^{\infty}\left|\gamma_n\right|^2 \leq \frac{1}{4\left(1+\alpha\right)^2}\left(\frac{\pi^2}{6}-2 \mathrm{Li}_2\left(-\alpha\right)+ \mathrm{Li}_2\left(\alpha^2\right)\right), \end{array}$$ where Li2 denotes the dilogarithm function are investigated.

scholarly journals Sufficiency Criteria for q -Starlike Functions Associated with Cardioid

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Saira Zainab ◽  
Ayesha Shakeel ◽  
Muhammad Imran ◽  
Nazeer Muhammad ◽  
Hira Naz ◽  
...  
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Image Domain ◽  
Lemniscate Of Bernoulli ◽  
Differential Subordinations

This article deals with the q -differential subordinations for starlike functions associated with the lemniscate of Bernoulli and cardioid domain. The primary goal of this work is to find the conditions on γ for 1 + γ z ∂ q   h z / h n   z   ≺ 1 + z , where h z is analytic function and is subordinated by the function which is producing cardioid domain as its image domain while mapping the open unit disk. Along with this, certain sufficient conditions for q -starlikeness of analytic functions are determined.

scholarly journals On Lie ideals and $ (\sigma, \tau)$-Jordan derivations An application of Briot-Bouquet differential subordination

2002 ◽  
Vol 33 (1) ◽  
pp. 1-12
Author(s):  
Jagannath Patel
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Function Theory ◽  
Differential Subordination ◽  
Open Unit ◽  
Geometric Function Theory ◽  
Open Unit Disc ◽  
Geometric Function

By using the method of Briot-Bouquet differential subordination, we prove and sharpen some classical results in geometric function theory. We also derive some criteria for univalency for certain classes analytic functions in the open unit disc.

scholarly journals Analytic Univalent Functions Defined by Generalized Discrete Probability Distribution

2020 ◽  
pp. 169-178
Author(s):  
Abiodun Tinuoye Oladipo
Keyword(s):  
Probability Distribution ◽  
Function Theory ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Geometric Function Theory ◽  
Strong Connection ◽  
Discrete Probability ◽  
Discrete Probability Distribution ◽  
Geometric Function ◽  
Coefficient Inequalities

The close-to-convex analogue of a starlike functions by means of generalized discrete probability distribution and Poisson distribution was considered. Some coefficient inequalities and their connection to classical Fekete-Szego theorem are obtained. Our results provide strong connection between Geometric Function Theory and Statistics.

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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