Coefficient Inequalities for Concave and Meromorphically Starlike Univalent Functions

TL;DR

This paper establishes coefficient inequalities for a class of meromorphic univalent functions with specific geometric properties, providing sharp bounds and extending the understanding of such functions' behavior.

Contribution

It introduces new coefficient estimates for concave and meromorphically starlike univalent functions, including sharp bounds, expanding the theoretical framework of geometric function theory.

Findings

Derived sharp coefficient bounds for the classes $Co(p)$ and $ ext{Sigma}^s(p,w_0)$.

Extended existing theories by characterizing the extremal functions achieving equality.

Provided new insights into the geometric properties of these 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.