Meromorphic functions with small Schwarzian derivative

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.