On harmonic Bloch-type mappings

TL;DR

This paper introduces a new class of harmonic mappings called Bloch-type functions, generalizing the analytic Bloch space, and provides estimates and characterizations for these functions.

Contribution

It defines a harmonic Bloch-type class based on Jacobian bounds and establishes analogues of classical theorems for this new class.

Findings

Introduces harmonic Bloch-type functions with Jacobian-based bounds

Provides estimates for schlicht radius, growth, and coefficients

Establishes an analogue of the Bloch theorem involving univalent functions

Abstract

Let $f$ be a complex-valued harmonic mapping defined in the unit disk $\mathbb D$. We introduce the following notion: we say that $f$ is a Bloch-type function if its Jacobian satisfies $$ \sup_{z\in\mathbb D}(1-|z|^2)\sqrt{|J_f(z)|}<\infty. $$ This gives rise to a new class of functions which generalizes and contains the well-known analytic Bloch space. We give estimates for the schlicht radius, the growth and the coefficients of functions in this class. We establish an analogue of the theorem which states that an analytic $\varphi$ is Bloch if and only if there exists $c>0$ and a univalent $\psi$ such that $\varphi = c \log \psi'$.