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Criteria for Bounded Valence of Harmonic Mappings

Criteria for Bounded Valence of Harmonic Mappings In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that $$\limsup _{|z|\rightarrow 1} |S(f)(z)| (1-|z|^2)^2 < 2$$ lim sup | z | → 1 | S ( f ) ( z ) | ( 1 - | z | 2 ) 2 < 2 , then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that $$\limsup _{|z|\rightarrow 1} |f''(z)/f'(z)|\, (1-|z|^2)< 1$$ lim sup | z | → 1 | f ′ ′ ( z ) / f ′ ( z ) | ( 1 - | z | 2 ) < 1 . In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Criteria for Bounded Valence of Harmonic Mappings

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References (16)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-017-0197-z
Publisher site
See Article on Publisher Site

Abstract

In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that $$\limsup _{|z|\rightarrow 1} |S(f)(z)| (1-|z|^2)^2 < 2$$ lim sup | z | → 1 | S ( f ) ( z ) | ( 1 - | z | 2 ) 2 < 2 , then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that $$\limsup _{|z|\rightarrow 1} |f''(z)/f'(z)|\, (1-|z|^2)< 1$$ lim sup | z | → 1 | f ′ ′ ( z ) / f ′ ( z ) | ( 1 - | z | 2 ) < 1 . In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic.

Journal

Computational Methods and Function Theory – Springer Journals

Published: Mar 28, 2017

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