Meromorphic functions with small Schwarzian derivative

Abstract

We consider the family of all meromorphic functions f of the form f(z) =1/z + b0 + b1z + b2z 2 + analytic and locally univalent in the puncture disk D0 := (z ∈ C: 0 < jzj < 1). Our first objective in this paper is to find a sufficient condition for f to be meromorphically convex of order α, 0 ≤ α < 1, in terms of the fact that the absolute value of the well-known Schwarzian derivative Sf (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 + a2z2 + a3z3 + analytic and locally univalent in the open unit disk D := (z ∈ C: |z| < 1g, and show that g is belonging to a family of functions convex in one direction if |Sg(z)| is bounded above by a small positive constant depending on the second coefficient a2. In particular, we show that such functions g are also contained in the starlike and close-to-convex family. © 2018 Babes-Bolyai University.