Rational Ahlfors Functions

TL;DR

This paper characterizes degree two rational Ahlfors functions by positive residues, demonstrates the characterization's limitations for higher degrees, and provides examples across all degrees using numerical methods.

Contribution

It proves that degree two rational Ahlfors functions are characterized by positive residues and shows this does not extend to higher degrees, with new examples provided.

Findings

Degree two rational Ahlfors functions have positive residues.

The positive residue characterization does not hold for degrees higher than two.

Examples of rational Ahlfors functions are constructed for all degrees.

Abstract

We study a problem of Jeong and Taniguchi asking to find all rational maps which are Ahlfors functions. We prove that the rational Ahlfors functions of degree two are characterized by having positive residues at their poles. We then show that this characterization does not generalize to higher degrees, with the help of a numerical method for the computation of analytic capacity. We also provide examples of rational Ahlfors functions in all degrees.