scholarly journals Realization of Simply Connected Polygonal Linkages and Recognition of Unit Disk Contact Trees

2015 ◽  
pp. 447-459 ◽  
Author(s):  
Clinton Bowen ◽  
Stephane Durocher ◽  
Maarten Löffler ◽  
Anika Rounds ◽  
André Schulz ◽  
...  
Keyword(s):  
Unit Disk ◽  
Disk Contact ◽  
Polygonal Linkages ◽  
Simply Connected

scholarly journals Neo-Normal Functions in Arbitrary Regions

1972 ◽  
Vol 48 ◽  
pp. 19-36
Author(s):  
Kam-Fook Tse
Keyword(s):  
Unit Disk ◽  
Meromorphic Functions ◽  
Boundary Behavior ◽  
Universal Covering ◽  
Value Distribution ◽  
Covering Surface ◽  
Normal Functions ◽  
Definition Of ◽  
Simply Connected Region ◽  
Simply Connected

It is well known that many properties possessed by functions holomorphic and bounded in a region are also possessed by functions meromorphic and omitting three values. Noshiro [14] in 1938 and Lehto and Virtanen [12] in 1957 independently defined the notion of “normal functions” ; they and many others subsequently discovered that most properties concerning boundary behavior and value distribution acquired by meromorphic functions omitting three values in the unit disk (or more general, in a simply-connected region) are also valid properties of “normal functions” defined there. In their research on the problems of value distribution of normal functions, Lange [9], Gavrilov [5] and Gauthier [4] have discovered that functions normal in the disk are exactly those which omit three values “locally,” i.e., they do not possess any “p-sequence” (see above references). However, the definition of a function being normal in a region depends on the simply-connectedness of the region or its universal covering surface. It is thus difficult to judge if a function defined in an arbitrary region is normal.

Extension Operators for Biholomorphic Mappings

2019 ◽  
Vol 62 (3) ◽  
pp. 671-679
Author(s):  
Jianfei Wang ◽  
Danli Zhang
Keyword(s):  
Unit Disk ◽  
Unbounded Domain ◽  
Geometric Approach ◽  
Starlike Function ◽  
New Method ◽  
Convex Mapping ◽  
Starlike Mapping ◽  
The Hyperbolic Metric ◽  
Image Position ◽  
Simply Connected

AbstractSuppose that $D\subset \mathbb{C}$ is a simply connected subdomain containing the origin and $f(z_{1})$ is a normalized convex (resp., starlike) function on $D$. Let $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{N}(D)=\bigg\{(z_{1},w_{1},\ldots ,w_{k})\in \mathbb{C}\times \mathbb{C}^{n_{1}}\times \cdots \times \mathbb{C}^{n_{k}}:\Vert w_{1}\Vert _{p_{1}}^{p_{1}}+\cdots +\Vert w_{k}\Vert _{p_{k}}^{p_{k}}<\frac{1}{\unicode[STIX]{x1D706}_{D}(z_{1})}\bigg\},\end{eqnarray}$$ where $p_{j}\geqslant 1$, $N=1+n_{1}+\cdots +n_{k}$, $w_{1}\in \mathbb{C}^{n_{1}},\ldots ,w_{k}\in \mathbb{C}^{n_{k}}$ and $\unicode[STIX]{x1D706}_{D}$ is the density of the hyperbolic metric on $D$. In this paper, we prove that $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)(z_{1},w_{1},\ldots ,w_{k})=(f(z_{1}),(f^{\prime }(z_{1}))^{1/p_{1}}w_{1},\ldots ,(f^{\prime }(z_{1}))^{1/p_{k}}w_{k})\end{eqnarray}$$ is a normalized convex (resp., starlike) mapping on $\unicode[STIX]{x1D6FA}_{N}(D)$. If $D$ is the unit disk, then our result reduces to Gong and Liu via a new method. Moreover, we give a new operator for convex mapping construction on an unbounded domain in $\mathbb{C}^{2}$. Using a geometric approach, we prove that $\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)$ is a spiral-like mapping of type $\unicode[STIX]{x1D6FC}$ when $f$ is a spiral-like function of type $\unicode[STIX]{x1D6FC}$ on the unit disk.

scholarly journals On some properties of solutions of the p-harmonic equation

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 577-591 ◽  
Author(s):  
Sh. Chen ◽  
S. Ponnusamy ◽  
X. Wang
Keyword(s):  
Unit Disk ◽  
Connected Domain ◽  
Harmonic Mappings ◽  
Simply Connected Domain ◽  
Harmonic Equation ◽  
Simply Connected ◽  
Complex Valued ◽  
Class Of Functions

A 2p-times continuously differentiable complex-valued function ? = u + iv in a simply connected domain ? ? C is p-harmonic if ? satisfies the p-harmonic equation ?p? = 0. In this paper, we investigate the properties of p-harmonic mappings in the unit disk |z| < 1. First, we discuss the convexity, the starlikeness and the region of variability of some classes of p-harmonic mappings. Then we prove the existence of Landau constant for the class of functions of the form D? = z?z - ??z, where f is p-harmonic in |z| < 1. Also, we discuss the region of variability for certain p-harmonic mappings. At the end, as a consequence of the earlier results of the authors, we present explicit upper estimates for Bloch norm for bi- and tri-harmonic mappings.

Means and coefficients of functions which omit a sequence of values

1977 ◽  
Vol 81 (1) ◽  
pp. 47-57
Author(s):  
Albert Baernstein ◽  
Richard Rochberg
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
Maximum Modulus ◽  
Distortion Theorem ◽  
Principle Of Subordination ◽  
Image Position ◽  
Simply Connected

Suppose that f is analytic in the unit disk D. If its range f(D) is contained in a simply connected proper subdomain of the plane, then the principle of subordination and the distortion theorem for univalent functions show thatwhere M(r, f) denotes the maximum modulus of f. Cartwright (2) studied functions which, instead of omitting all values on a continuum stretching to infinity, omit only a sequence of values. She assumed that the sequence {wn} satisfiesandand proved that if f(D) contains none of the points {wn} thenmfor every ε > 0. Cartwright's proof was based on the Ahlfors Distortion Theorem, and is quite complicated. A much simpler proof was given by Pommerenke in (10). The key idea in his proof will also be used in the present paper.

scholarly journals Convexity properties of the normalized Steklov zeta function of a planar domain

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexandre Jollivet
Keyword(s):  
Zeta Function ◽  
Unit Disk ◽  
Elementary Proof ◽  
Closed Curve ◽  
Planar Domain ◽  
Closed Unit Disk ◽  
Convexity Properties ◽  
The Difference ◽  
Simply Connected ◽  
Dirichlet To Neumann Operator

AbstractWe consider the zeta function \zeta_{\Omega} for the Dirichlet-to-Neumann operator of a simply connected planar domain Ω bounded by a smooth closed curve of perimeter 2\pi. We name the difference \zeta_{\Omega}-\zeta_{\mathbb{D}} the normalized Steklov zeta function of the domain Ω, where 𝔻 denotes the closed unit disk. We prove that (\zeta_{\Omega}-\zeta_{\mathbb{D}})^{\prime\prime}(0)\geq 0 with equality if and only if Ω is a disk. We also provide an elementary proof that, for a fixed real 𝑠 satisfying s\leq-1, the estimate (\zeta_{\Omega}-\zeta_{\mathbb{D}})^{\prime\prime}(s)\geq 0 holds with equality if and only if Ω is a disk. We then bring examples of domains Ω close to the unit disk where this estimate fails to be extended to the interval (0,2). Other computations related to previous works are also detailed in the remaining part of the text.

The Yukawa Potential Function and Reciprocal Univalent Function

2013 ◽  
Vol 3 (2) ◽  
pp. 197-202
Author(s):  
Amir Pishkoo ◽  
Maslina Darus
Keyword(s):  
Mathematical Model ◽  
Unit Disk ◽  
Potential Function ◽  
Univalent Function ◽  
Open Problem ◽  
Yukawa Potential ◽  
Reciprocal Theorem ◽  
Problem Statement ◽  
Fundamental Forces

This paper presents a mathematical model that provides analytic connection between four fundamental forces (interactions), by using modified reciprocal theorem,derived in the paper, as a convenient template. The essential premise of this work is to demonstrate that if we obtain with a form of the Yukawa potential function [as a meromorphic univalent function], we may eventually obtain the Coloumb Potential as a univalent function outside of the unit disk. Finally, we introduce the new problem statement about assigning Meijer's G-functions to Yukawa and Coloumb potentials as an open problem.

On a Subclass of Univalent Harmonic Mappings Convex in the Imaginary Direction

2020 ◽  
pp. 445-454
Author(s):  
Deepali Khurana ◽  
Sushma Gupta ◽  
Sukhjit Singh
Keyword(s):  
Present Article ◽  
Unit Disk ◽  
Imaginary Axis ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Distortion Bounds ◽  
Univalent Harmonic Mappings

In the present article, we consider a class of univalent harmonic mappings, $\mathcal{C}_{T} = \left\{ T_{c}[f] =\frac{f+czf'}{1+c}+\overline{\frac{f-czf'}{1+c}}; \; c>0\;\right\}$ and $f$ is convex univalent in $\mathbb{D}$, whose functions map the open unit disk $\mathbb{D}$ onto a domain convex in the direction of the imaginary axis. We estimate coefficient, growth and distortion bounds for the functions of the same class.

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