Document Type: Research Paper

Authors

1 Department of Mathematics‎, ‎Technical and Vocational‎, ‎University (TVU)‎, ‎Tehran‎, ‎Iran

2 Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran

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 $1<\beta\leq 2$‎, ‎and‎ ‎consisting of harmonic univalent function $f = h+\bar{g}$‎, ‎where $h$ and $g$ are in the form‎ ‎$h(z) = z+\sum\limits_{n=2}^\infty |a_n|z^n‎$ ‎and ‎‎$‎g(z) =‎\sum\limits_{n=2}^\infty |b_n|\bar z^n$‎ 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.

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