Landau theorem for planar harmonic mappings
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 \(\lambda_{\omega}(z)=||f_z(z)|-|f_{\bar{z}}(z)||\).
scientific article; zbMATH DE number 6506177