A note on convexity of convolutions of harmonic mappings
Bull. Korean Math. Soc. 2015 Vol. 52, No. 6, 1925-1935
https://doi.org/10.4134/BKMS.2015.52.6.1925
Published online November 30, 2015
Yue-Ping Jiang, Antti Rasila, and Yong Sun
Hunan University, Aalto University, Hunan University
Abstract : In this paper, we study right half-plane harmonic mappings $f_0$ and $f$, where $f_0$ is fixed and $f$ is such that its dilatation of a conformal automorphism of the unit disk. We obtain a sufficient condition for the convolution of such mappings to be convex in the direction of the real axis. The result of the paper is a generalization of the result of by Li and Ponnusamy \cite{lp1}, which itself originates from a problem posed by Dorff {\it et al.} in \cite{dor2012}.
Keywords : harmonic univalent mapping, convolution, half-plane mapping, convex function
MSC numbers : Primary 30C45; Secondary 30C20, 30C65
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