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.