A new subclass of harmonic mappings with positive coefficients

Subject Areas : Complex Analysis

A. R. Haghighi ¹ , N. Asghary ² , A. Sedghi ³

1 - Department of Mathematics, Technical and Vocational, University (TVU), Tehran, Iran
2 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran
3 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran

Received: 2018-10-16 Accepted : 2019-01-12 Published : 2019-08-01

Keywords: Convex combinations&lrm, , &lrm, extreme points&lrm, , &lrm, harmonic starlike functions&lrm, , &lrm, harmonic univalent functions,

Abstract :

‎Complex-valued harmonic functions that are univalent and‎‎sense-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 $1<\beta\leq 2$‎, ‎and‎‎consisting 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 under‎convolution and convex combinations‎.

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