Landau theorem for planar harmonic mappings
type: Item
instance of: scholarly article
title: Landau theorem for planar harmonic mappings
zbMATH Open document ID: 1329.31002
author: Jianfeng Zhu
published in: Complex Analysis and Operator Theory
publication date: 2015-11-09T00:00:00Z
review text: This paper is about estimates on Bloch constants for planar harmonic mappings. Roughly speaking, Bloch constants show to what extent a holomorphic function on the unit disk \(\mathbb{D}\) can be univalent. To be precise, let \(B_f\) be the least upper bound of all numbers \(b>0\) such that \(f\) maps a region \(\Omega\in \mathbb{D}\) univalently onto \(\{z\in \mathbb{C}:|z-z_0|<b\}\). The Bloch constant B is defined as \(B=\inf\{B_f:f'(0)=1\}\), where \(f\) is a holomorphic function on \(\mathbb{D}\). Some estimates for B have been already attained, but its exact value is still unknown. It is also known that, if \(f\) is bounded by a constant M, then by the classical Landau theorem, \(f\) is univalent in \(|z|<\rho_0=\frac{1}{M+\sqrt{M^2-1}}\) and \(f(|z|<\rho_0)\) contains a disk \(|\omega|<\sigma_0=M\rho_0^2\). The author improves the former results by some better estimates on the Bloch constants for bounded planar harmonic mappings, by computation, using the Poisson formula. By the Poisson formula, the harmonic mapping can be expressed as the integral of the Poisson kernel and the boundary function. So, the derivative can also be expressed by some integral of the boundary function. By approximation and computation, the author obtains that \[ |f(z_1)-f(z_2)|\geq |z_1-z_2|\bigg(1-\frac{2Mr\sqrt{2-r^2}}{1-r^2}\bigg) \] when \(|z_1|<r\) and \(|z_2|<r\). So, when \(1-\frac{2Mr\sqrt{2-r^2}}{1-r^2}>0\), that is, when \(r<r_0=\sqrt{1-\frac{2M}{\sqrt{1+4M^2}}}\), then \(f(z_1)\neq f(z_2)\). Also by computation, the author obtains some bounds for \(\Lambda_{\omega}(z)=|f_z(z)|+|f_{\bar{z}}(z)|\) and \(\lambda_{\omega}(z)=||f_z(z)|-|f_{\bar{z}}(z)||\).
reviewed by: Chong Wu
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 30C62
zbMATH DE Number: 6506177
zbMATH Keywords: Landau theorem
zbMATH Keywords: harmonic mapping
zbMATH Keywords: Bloch constant
zbMATH Keywords: Poisson formula
MaRDI profile type: Publication
full work available at URL: s11785-015-0449-8
OpenAlex ID: W2030758704
cites work: Über die Blochsche Konstante. (On the Bloch constant)
cites work: On Bloch's Constant
cites work: On Bloch's constant
cites work: Bloch constants for planar harmonic mappings
cites work: Q5618011
cites work: Q4828548
cites work: Radius of close-to-convexity and fully starlikeness of harmonic mappings
cites work: Estimates on Bloch constants for planar harmonic mappings
cites work: A variant of Schwarz-Pick lemma for planar harmonic mappings
cites work: Some estimates for harmonic mappings with given boundary function
DOI: 10.1007/S11785-015-0449-8
description: scientific article; zbMATH DE number 6506177