A new subclass of harmonic mappings with positive coefficients
محورهای موضوعی : Complex AnalysisA. 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
کلید واژه:
چکیده مقاله :
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.
