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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