Value distribution of functions regular in the unit disk

1983 ◽  
pp. 13-43
Author(s):  
W. K. Hayman
Keyword(s):  
Unit Disk ◽  
Value Distribution

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.

scholarly journals Some Results on Value Distribution of Meromorphic Functions in the Unit Disk

1969 ◽  
Vol 34 ◽  
pp. 105-119 ◽  
Author(s):  
Kam-Fook Tse
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Euclidean Distance ◽  
Meromorphic Functions ◽  
Riemann Sphere ◽  
Value Distribution ◽  
Open Unit ◽  
Open Unit Disk ◽  
Chordal Distance

Let C and D be the unit circle and the open unit disk respectively. We shall use p(z,z′) to represent the non-Euclidean distance [3, p. 263] between the two points z and z′ in D, and X(w, w′) to represent the chordal distance between the two points w and w′ on the Riemann Sphere Ω.

scholarly journals Unbounded functions in the unit disk

1983 ◽  
Vol 6 (2) ◽  
pp. 201-242 ◽  
Author(s):  
L. R. Sons
Keyword(s):  
Unit Disk ◽  
Unit Disc ◽  
Specific Growth ◽  
Value Distribution ◽  
Growth Indicator

A survey is made of results related to the value distribution of functions which are meromorphic or analytic in the unit disc and have unbounded growth according to some specific growth indicator.

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.

Value distribution and uniqueness of $q$-shift difference polynomials

2016 ◽  
Vol 46 (2) ◽  
pp. 33-44
Author(s):  
Pulak Sahoo ◽  
Gurudas Biswas
Keyword(s):  
Value Distribution ◽  
Difference Polynomials

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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