A new subclass of harmonic mappings with positive coefficients

الموضوعات :

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

تاريخ الإرسال : 07 الثلاثاء , صفر, 1440 تاريخ التأكيد : 06 السبت , جمادى الأولى, 1440 تاريخ الإصدار : 29 الخميس , ذو القعدة, 1440

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

ملخص المقالة :

‎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.

المصادر:

[1] J. Clunie, T. Sheil-Small, Univalent functions, Ann. Acad. Sci. Fenn. Series A. 9 (1984), 3-25.

[2] K. K. Dixit, S. Porwal, A subclass of harmonic univalent functions with positive coefficients, Tamk. J. Math. 41 (3) (2010), 261-269.

[3] A. R. Haghighi, A. Sadeghi, N. Asghary, A subclass of harmonic univalent functions, Acta Univ. Apul. 38 (2014), 1-10.

[4] S. Y. Karpuzogullari, M. Özturk, M. Y. Karadeniz, A subclass of harmonic univalent functions with negative coefficients, App. Math. Comp. 142 (2003), 469-476.

[5] St. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81 (1981), 521-528.

[6] H. Silverman, Univalent functions with negative coefficients, J. Math. Anal. App. 220 (1998), 283-289.