Meromorphic functions with small Schwarzian derivative

TL;DR

This paper establishes conditions involving the Schwarzian derivative for meromorphic functions to be convex of a certain order and explores the geometric properties of functions with bounded Schwarzian derivatives.

Contribution

It provides new sufficient conditions based on Schwarzian derivative bounds for meromorphic functions to be convex of order b1b0, and characterizes functions with bounded Schwarzian derivative as convex in one direction, starlike, and close-to-convex.

Findings

Bound on Schwarzian derivative ensures meromorphic convexity of order b1b0.

Functions with small Schwarzian derivative are convex in one direction.

Such functions are also starlike and close-to-convex.

Abstract

We consider the family of all meromorphic functions $f$ of the form $$ f(z)=\frac{1}{z}+b_0+b_1z+b_2z^2+\cdots $$ analytic and locally univalent in the puncture disk $\mathbb{D}_0:=\{z\in\mathbb{C}:\,0<|z|<1\}$. Our first objective in this paper is to find a sufficient condition for $f$ to be meromorphically convex of order $\alpha$, $0\le \alpha<1$, in terms of the fact that the absolute value of the well-known Schwarzian derivative $S_f (z)$ of $f$ is bounded above by a smallest positive root of a non-linear equation. Secondly, we consider a family of functions $g$ of the form $g(z)=z+a_2z^2+a_3z^3+\cdots$ analytic and locally univalent in the open unit disk $\mathbb{D}:=\{z\in\mathbb{C}:\,|z|<1\}$, and show that $g$ is belonging to a family of functions convex in one direction if $|S_g(z)|$ is bounded above by a small positive constant depending on the second coefficient $a_2$. In particular, we show that such functions $g$ are also contained in the starlike and close-to-convex family.