On the Analytic Part of Univalent Harmonic Mappings

Abstract

© 2018, Springer International Publishing AG, part of Springer Nature. In this article we obtain two sharp results concerning the analytic part of harmonic mappings f= h+ g¯ from the class SH0(S) which was recently introduced by Ponnusamy and Sairam Kaliraj. For example, we get the sharp estimate for | arg h′(z) | in the case when |z|≤1/2 and obtain the sharp radius of convexity for h. Our approach is applicable to a more general situation. Finally, we determine simple condition on the analytic part of univalent harmonic mappings so that it is in Hpspaces for 0 < p< 1 / 3.