The Schwarzian derivative and the Wiman-Valiron property

TL;DR

This paper investigates the properties of the Schwarzian derivative for certain transcendental meromorphic functions, showing it lacks specific types of transcendental singularities and exceptional values under given conditions.

Contribution

It demonstrates that for functions with finitely many critical values and bounded multiplicities, the Schwarzian derivative has no direct transcendental singularity over infinity.

Findings

Schwarzian derivative lacks a direct transcendental singularity over infinity.

It does not have infinity as a Borel exceptional value.

Results apply to functions with finitely many critical values and bounded multiplicities.

Abstract

Suppose that a transcendental meromorphic function in the plane has finitely many critical values, while its multiple points have bounded multiplicities, and its inverse function has finitely many transcendental singularities. Using the Wiman-Valiron method it is shown that the Schwarzian derivative does not have a direct transcendental singularity over infinity, and does not have infinity as a Borel exceptional value.