Coefficient Inequalities for Concave and Meromorphically Starlike Univalent Functions

Abstract

Let $\ID$ denote the open unit disk and $f:\,\ID\TO\BAR\IC$ be meromorphic and univalent in $\ID$ with the simple pole at $p\in (0,1)$ and satisfying the standard normalization $f(0)=f'(0)-1=0$. Also, let $f$ have the expansion $$f(z)=\sum_{n=-1}^{\infty}a_n(z-p)^n,\quad |z-p|<1-p, $$ such that $f$ maps $\ID$ onto a domain whose complement with respect to $\BAR{\IC}$ is a convex set (starlike set with respect to a point $w_0\in \IC, w_0\neq 0$ resp.). We call these functions as concave (meromorphically starlike resp.) univalent functions and denote this class by $Co(p)$ $(\Sigma^s(p, w_0)$ resp.). We prove some coefficient estimates for functions in the classes where the sharpness of these estimates is also achieved.