On some problems of James Miller

TL;DR

This paper studies a class of meromorphic univalent functions with a pole inside the unit disk, characterizes the variability of their second Taylor coefficient, and corrects previous results by James Miller.

Contribution

It determines the exact region of variability for the second Taylor coefficient in the class and rectifies earlier findings by James Miller.

Findings

Exact variability region for the second Taylor coefficient

Correction of previous results by James Miller

Characterization of meromorphic univalent functions with specific geometric properties

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 $w_0 \neq 0,\, \infty$. We denote this class of functions by $\Sigma^*(p,w_0)$. In this paper, we find the exact region of variability for the second Taylor coefficient for functions in $\Sigma^*(p,w_0)$. In view of this result we rectify some results of James Miller.