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A Valency Criterion for Harmonic Mappings

A Valency Criterion for Harmonic Mappings Let $$f=h+{\overline{g}}$$ f = h + g ¯ be a sense-preserving harmonic mapping of the closed unit disk $$\overline{{\mathbb {D}}}$$ D ¯ with a Blashke product dilatation $$B_m= g'/h'$$ B m = g ′ / h ′ of order m. The aim of this paper is to prove that if $$h'$$ h ′ has $$p-1$$ p - 1 zeros, counting multiplicity, in $${\mathbb {D}}$$ D and no zeros on $$\partial {\mathbb {D}},$$ ∂ D , and that $$\begin{aligned} {\text {Re}}\left\{ 1+\mathrm{{e}}^\mathrm{it}\frac{h''(\mathrm{{e}}^\mathrm{it})}{h'(\mathrm{{e}}^\mathrm{it})}\right\} >-\frac{1}{2}\sum _{k=1}^m\frac{1-|a_k|}{1+|a_k|}, \end{aligned}$$ Re 1 + e it h ′ ′ ( e it ) h ′ ( e it ) > - 1 2 ∑ k = 1 m 1 - | a k | 1 + | a k | , where $$a_1,\ldots , a_m$$ a 1 , … , a m are the zeros of $$B_m,$$ B m , then f is $$(m+p-1)$$ ( m + p - 1 ) -valent. The proof deploys a surface-theoretic technique based on an effective “pasting” procedure. This is an improvement of an earlier result of Bshouty et al. (Proc Am Math Soc 146:1113–1121, 2018) which asserts that if f is a sense-preserving harmonic mapping on $${\mathbb {D}},$$ D , with dilatation $$z^m$$ z m that satisfies the inequality $$\begin{aligned} {\text {Re}}\left\{ 1+ z\frac{h''(z)}{h'(z)}\right\} >-\frac{m}{2},\quad z\in {\mathbb {D}}, \end{aligned}$$ Re 1 + z h ′ ′ ( z ) h ′ ( z ) > - m 2 , z ∈ D , then f is $$(m+p)$$ ( m + p ) -valent. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

A Valency Criterion for Harmonic Mappings

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-019-00274-z
Publisher site
See Article on Publisher Site

Abstract

Let $$f=h+{\overline{g}}$$ f = h + g ¯ be a sense-preserving harmonic mapping of the closed unit disk $$\overline{{\mathbb {D}}}$$ D ¯ with a Blashke product dilatation $$B_m= g'/h'$$ B m = g ′ / h ′ of order m. The aim of this paper is to prove that if $$h'$$ h ′ has $$p-1$$ p - 1 zeros, counting multiplicity, in $${\mathbb {D}}$$ D and no zeros on $$\partial {\mathbb {D}},$$ ∂ D , and that $$\begin{aligned} {\text {Re}}\left\{ 1+\mathrm{{e}}^\mathrm{it}\frac{h''(\mathrm{{e}}^\mathrm{it})}{h'(\mathrm{{e}}^\mathrm{it})}\right\} >-\frac{1}{2}\sum _{k=1}^m\frac{1-|a_k|}{1+|a_k|}, \end{aligned}$$ Re 1 + e it h ′ ′ ( e it ) h ′ ( e it ) > - 1 2 ∑ k = 1 m 1 - | a k | 1 + | a k | , where $$a_1,\ldots , a_m$$ a 1 , … , a m are the zeros of $$B_m,$$ B m , then f is $$(m+p-1)$$ ( m + p - 1 ) -valent. The proof deploys a surface-theoretic technique based on an effective “pasting” procedure. This is an improvement of an earlier result of Bshouty et al. (Proc Am Math Soc 146:1113–1121, 2018) which asserts that if f is a sense-preserving harmonic mapping on $${\mathbb {D}},$$ D , with dilatation $$z^m$$ z m that satisfies the inequality $$\begin{aligned} {\text {Re}}\left\{ 1+ z\frac{h''(z)}{h'(z)}\right\} >-\frac{m}{2},\quad z\in {\mathbb {D}}, \end{aligned}$$ Re 1 + z h ′ ′ ( z ) h ′ ( z ) > - m 2 , z ∈ D , then f is $$(m+p)$$ ( m + p ) -valent.

Journal

Computational Methods and Function TheorySpringer Journals

Published: May 25, 2019

References