A new subclass of harmonic mappings with positive coefficients

Abstract

‎Complex-valued harmonic functions that are univalent and‎ ‎sense-preserving in the open unit disk $U$ can be written as form‎ ‎$f =h+bar{g}$‎, ‎where $h$ and $g$ are analytic in $U$‎. ‎In this paper‎, ‎we introduce the class $S_H^1(beta)$‎, ‎where $1<betaleq 2$‎, ‎and‎ ‎consisting of harmonic univalent function $f = h+bar{g}$‎, ‎where $h$ and $g$ are in the form‎ ‎$h(z) = z+sumlimits_{n=2}^infty |a_n|z^n‎$ ‎and ‎‎$‎g(z) =‎sumlimits_{n=2}^infty |b_n|bar z^n$‎ for which‎ ‎$$mathrm{Re}left{z^2(h''(z)+g''(z))‎ +2z(h'(z)+g'(z))-(h(z)+g(z))-(z-1)right}<beta.$$‎ It is shown that the members of this class are convex and starlike‎. ‎We obtain distortions bounds extreme point for functions belonging to this class‎, ‎and we also show this class is closed under‎ convolution and convex combinations‎.