Hyperbolic linear invariance and hyperbolic k-convexity

AbstractPommerenke initiated the study of linearly invariant families of locally schlicht holomorphic functions defined on the unit disk The concept of linear invariance has proved fruitful in geometric function theory. One aspect of Pommerenke's work is the extension of certain results from classical univalent function theory to linearly invariant functions. We propose a definition of a related concept that we call hyperbolic linear invariance for locally schlicht holomorphic functions that map the unit disk into itself. We obtain results for hyperbolic linearly invariant functions which generalize parts of the theory of bounded univalent functions. There are many similarities between linearly invariant functions and hyperbolic linearly invariant functions, but some new phenomena also arise in the study of hyperbolic linearly invariant functions.

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On a new linear operator formulated by Airy functions in the open unit disk

Advances in Difference Equations ◽

10.1186/s13662-021-03527-1 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Rabha W. Ibrahim ◽

Dumitru Baleanu

Keyword(s):

Linear Operator ◽

Unit Disk ◽

Function Theory ◽

Univalent Functions ◽

Open Unit ◽

Geometric Function Theory ◽

Open Unit Disk ◽

Airy Functions ◽

Geometric Function ◽

Difference Formula

AbstractIn this note, we formulate a new linear operator given by Airy functions of the first type in a complex domain. We aim to study the operator in view of geometric function theory based on the subordination and superordination concepts. The new operator is suggested to define a class of normalized functions (the class of univalent functions) calling the Airy difference formula. As a result, the suggested difference formula joining the linear operator is modified to different classes of analytic functions in the open unit disk.

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Initial Coefficient Estimates and Fekete–Szegö Inequalities for New Families of Bi-Univalent Functions Governed by (p − q)-Wanas Operator

Symmetry ◽

10.3390/sym13112118 ◽

2021 ◽

Vol 13 (11) ◽

pp. 2118

Author(s):

Abbas Kareem Wanas ◽

Luminiţa-Ioana Cotîrlǎ

Keyword(s):

Present Article ◽

Unit Disk ◽

Function Theory ◽

Univalent Functions ◽

Upper Bounds ◽

Geometric Function Theory ◽

Coefficient Estimates ◽

Quantum Calculus ◽

Initial Coefficient ◽

Geometric Function

The motivation of the present article is to define the (p−q)-Wanas operator in geometric function theory by the symmetric nature of quantum calculus. We also initiate and explore certain new families of holormorphic and bi-univalent functions AE(λ,σ,δ,s,t,p,q;ϑ) and SE(μ,γ,σ,δ,s,t,p,q;ϑ) which are defined in the unit disk U associated with the (p−q)-Wanas operator. The upper bounds for the initial Taylor–Maclaurin coefficients and Fekete–Szegö-type inequalities for the functions in these families are obtained. Furthermore, several consequences of our results are pointed out based on the various special choices of the involved parameters.

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Analytic Univalent Functions Defined by Generalized Discrete Probability Distribution

Earthline Journal of Mathematical Sciences ◽

10.34198/ejms.5121.169178 ◽

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.

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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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