Coefficient inequalities for concave and meromorphically starlike univalent functions
Volume 93 / 2008
Abstract
Let $\mathbb D$ denote the open unit disk and $f:\mathbb D\rightarrow\overline{\mathbb C}$ be meromorphic and univalent in $\mathbb D$ with a simple pole at $p\in (0,1)$ and satisfying the standard normalization $f(0)=f'(0)-1=0$. Also, assume that $f$ has the expansion $$ f(z)=\sum_{n=-1}^{\infty}a_n(z-p)^n,\quad |z-p|<1-p, $$ and maps $\mathbb D$ onto a domain whose complement with respect to $\overline{\mathbb C}$ is a convex set (starlike set with respect to a point $w_0\in \mathbb C, w_0\neq 0$ resp.). We call such functions concave (meromorphically starlike resp.) univalent functions and denote this class by ${\rm Co}(p)$ $({\mit\Sigma}^{\rm s}(p, w_0)$ resp.). We prove some coefficient estimates for functions in these classes; the sharpness of these estimates is also established.
