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Rational Ahlfors Functions

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.