Sharp estimate on univalent radius for planar harmonic mappings with bounded Fréchet derivative

Abstract

Given harmonic mappings <italic>f</italic>(<italic>z</italic>) = <italic>h</italic>(<italic>z</italic>) + <italic>g</italic>(<italic>z</italic>) on the unit disk <italic>D</italic> ={<italic>z</italic>||<italic>z</italic>|&lt; 1}, where <italic>h</italic>(<italic>z</italic>) and <italic>g</italic>(<italic>z</italic>) are analytic functions on the unit disk <italic>D</italic>, with <italic>f</italic>(0) = 0, λ_{<italic>f</italic>} (0) = 1 and Λ_{<italic>f</italic>}≤Λ, by introducing one complex parameter λ, we consider the properties for the harmonic mappings <italic>F</italic>_λ(<italic>z</italic>) = <italic>h</italic>(<italic>z</italic>) + λ<italic>g</italic>(<italic>z</italic>) and analytic functions <italic>G</italic>_λ(<italic>z</italic>) = <italic>h</italic>(<italic>z</italic>)+ λ<italic>g</italic>(<italic>z</italic>) with |λ|= 1 and obtain the sharp estimate on univalent radius for <italic>F</italic>_λ(<italic>z</italic>) and <italic>G</italic>_λ(<italic>z</italic>). As an application, we also obtain a better estimate on Bloch constant for some <italic>K</italic>-quasiregular harmonic mappings on the unit disk <italic>D</italic>. Our results generalize and improve the one made by Chen et al. (2000).