On some problems of James Miller

Abstract

We consider the class of meromorphic univalent functions having a simple pole at $p\in (0,1)$ and that map the unit disc onto the exterior of a domain which is starlike with respect to a point $w_0\ne 0,\infty$. We denote this class of functions by ${\Sigma }^{\ast }(p,w_0)$. In this paper, we find the exact region of variability for the second Taylor coefficient for functions in ${\Sigma }^{\ast }(p,w_0)$. In view of this result we rectify some results of James Miller.