scholarly journals Boundary characteristics of meromorphic functions with summable spherical derivation and annular functions. Consideration

2021 ◽  
Vol 51 ◽  
pp. 5-17
Author(s):  
Žarko Pavićević ◽  
Valerian Ivanovich Gavrilov
Keyword(s):  
Unit Circle ◽  
Meromorphic Functions ◽  
Boundary Behavior ◽  
Holomorphic Functions ◽  
Spherical Derivative ◽  
Boundary Properties ◽  
Boundary Characteristics ◽  
Annular Functions

In this paper we formulate classical theorems Plesner and Meyer on the boundary behavior of meromorphic functions and their refinement and strengthening - Gavrilov's and Kanatnikov's theorems. An application of these theorems to classes of meromorphic functions with integrable spherical derivative and annular holomorphic functions is presented. Collingwood's theorem on boundary singularities of the Tsuji function as well as Kanatnikov's theorems are formulated. Kanatnikov's theorems strengthen and generalize Collingwood's theorem to broader classes of meromorphic functions with summable spherical derivatives. Special attention is paid to the boundary properties of annular holomorphic functions. The behavior of annular holomorphic functions on the boundary of the unit circle is considered. It is shown that Gavrilov's P-sequences play an important role in the study of the boundary properties of holomorphic and meromorphic functions.

scholarly journals On the boundary behavior of holomorphic functions in the unit disc

1976 ◽  
Vol 21 (1) ◽  
pp. 36-48 ◽  
Author(s):  
Hidenobu Yoshida
Keyword(s):  
Unit Disc ◽  
Boundary Behavior ◽  
Holomorphic Functions ◽  
Fundamental Result ◽  
Boundary Properties

Seidel (1959) established various boundary properties of holomorphic functions with spiral asymptotic paths. Especially, Theorem 4 which is the fundamental result of the paper; was generalized by Faust (1962), using essentially the same method that was employed by Seidel.

scholarly journals A Boundary Theorem for Tsuji Functions

1967 ◽  
Vol 29 ◽  
pp. 197-200 ◽  
Author(s):  
E. F. Collingwood
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Spherical Derivative

Let D denote the unit disc | z | <1, C the unit circle | z | = 1 and Cr the circle | z| = r. Corresponding to any function w(z) meromorphic in D we denote by w*(z) the spherical derivative

scholarly journals A note on the phase retrieval of holomorphic functions

2020 ◽  
pp. 1-8
Author(s):  
Rolando Perez
Keyword(s):  
Unit Circle ◽  
Connected Domain ◽  
Unit Disc ◽  
Phase Retrieval ◽  
Holomorphic Functions ◽  
Nevanlinna Class

Abstract We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then $f=g$ up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of $\pi $ . We also prove that if f and g are functions in the Nevanlinna class, and if $|f|=|g|$ on the unit circle and on a circle inside the unit disc, then $f=g$ up to the multiplication of a unimodular constant.

Boundary Behavior and Quasi-Normality of Finitely Valent Holomorphic Functions

1973 ◽  
Vol 25 (4) ◽  
pp. 812-819
Author(s):  
David C. Haddad
Keyword(s):  
Complex Number ◽  
Unit Disc ◽  
Boundary Behavior ◽  
Dense Subset ◽  
Holomorphic Functions ◽  
Asymptotic Values

A function denned in a domain D is n-valent in D if f(z) — w0 has at most n zeros in D for each complex number w0. Let denote the class of nonconstant, holomorphic functions f in the unit disc that are n-valent in each component of the set . MacLane's class is the class of nonconstant, holomorphic functions in the unit disc that have asymptotic values at a dense subset of |z| = 1.

Local Boundary Behavior of Bounded Holomorphic Functions

1978 ◽  
Vol 30 (03) ◽  
pp. 583-592 ◽  
Author(s):  
Alexander Nagel ◽  
Walter Rudin
Keyword(s):  
Smooth Boundary ◽  
Boundary Behavior ◽  
Holomorphic Functions ◽  
Area Measure ◽  
Surface Area Measure ◽  
Smooth Curves ◽  
Local Boundary ◽  
Bounded Holomorphic Function ◽  
Bounded Holomorphic ◽  
Dimensional Lebesgue Measure

Let D ⊂⊂ Cn be a bounded domain with smooth boundary ∂D, and let F be a bounded holomorphic function on D. A generalization of the classical theorem of Fatou says that the set E of points on ∂D at which F fails to have nontangential limits satisfies the condition σ (E) = 0, where a denotes surface area measure. We show in the present paper that this result remains true when σ is replaced by 1-dimensional Lebesgue measure on certain smooth curves γ in ∂D. The condition that γ must satisfy is that its tangents avoid certain directions.

scholarly journals Some Remarks on Boundary Behavior of Analytic and Meromorphic Functions

1955 ◽  
Vol 9 ◽  
pp. 79-85 ◽  
Author(s):  
F. Bagemihl ◽  
W. Seidel
Keyword(s):  
Meromorphic Function ◽  
Uniqueness Theorem ◽  
Meromorphic Functions ◽  
Boundary Behavior ◽  
Regular Function ◽  
Simple Manner ◽  
Regular Functions ◽  
Almost All ◽  
Jordan Arcs

This paper is concerned with regular and meromorphic functions in |z| < 1 and their behavior near |z| = 1. Among the results obtained are the following. In section 2 we prove the existence of a non-constant meromorphic function that tends to zero at every point of |z| = 1 along almost all chords of |z| < 1 terminating in that point. Section 3 deals with the impossibility of ex tending this result to regular functions. In section 4 it is shown that a regular function can tend to infinity along every member of a set of spirals approach ing |z| = 1 and exhausting |z| < 1 in a simple manner. Finally, in section 5 we prove that this set of spirals cannot be replaced by an exhaustive set of Jordan arcs terminating in points of |z| = 1; Theorem 3 of this section can be interpreted as a uniqueness theorem for meromorphic functions.

scholarly journals On a problem of Bonar concerning Fatou points for annular functions

1975 ◽  
Vol 56 ◽  
pp. 163-170
Author(s):  
Akio Osada
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Open Unit ◽  
Open Unit Disk ◽  
Minimum Modulus ◽  
Jordan Curves ◽  
Fatou Points ◽  
Annular Functions

The purpose of this paper is to study the distribution of Fatou points of annular functions introduced by Bagemihl and Erdös [1]. Recall that a function f(z), regular in the open unit disk D: | z | < 1, is referred to as an annular function if there exists a sequence {Jn} of closed Jordan curves, converging out to the unit circle C: | z | = 1, such that the minimum modulus of f(z) on Jn increases to infinity. If the Jn can be taken as circles concentric with C, f(z) will be called strongly annular.

Sign in / Sign up

Export Citation Format

Share Document