Domains of variability of Laurent coefficients and the convex hull for the family of concave univalent functions

TL;DR

This paper investigates the variability of Laurent coefficients for a family of meromorphic functions with convex complement images, determining exact coefficient domains and disproving a related convex hull conjecture.

Contribution

It precisely characterizes the coefficient domains for the family of concave univalent functions and refutes a conjecture about their convex hull for specific parameters.

Findings

Exact domains of Laurent coefficients determined

Disproof of a conjecture on the convex hull of the family

Analysis of coefficient variability for specific parameter ranges

Abstract

Let $\ID$ denote the open unit disc and let $p\in (0,1)$. We consider the family $Co(p)$ of functions $f:\ID\to \overline{\IC}$ that satisfy the following conditions: \bee \item[(i)] $f$ is meromorphic in $\ID$ and has a simple pole at the point $p$. \item[(ii)] $f(0)=f'(0)-1=0$. \item[(iii)] $f$ maps $\ID$ conformally onto a set whose complement with respect to $\overline{\IC}$ is convex. \eee We determine the exact domains of variability of some coefficients $a_n(f)$ of the Laurent expansion $$f(z)=\sum_{n=-1}^{\infty} a_n(f)(z-p)^n,\quad |z-p|<1-p, $$ for $f\in Co(p)$ and certain values of $p$. Knowledge on these Laurent coefficients is used to disprove a conjecture of the third author on the closed convex hull of $Co(p)$ for certain values of $p$.