On a problem of Polya and Szegö

Abstract

We give a new proof of a theorem, which is originally due to Gehring and Pommerenke on the triviality of the extrema set Mf of the inner mapping radius |f′(ζ)|(1 - |ζ|2) over the unit disk in the plane, where the Riemann mapping function f satisfies the well-known Nehari univalence criterion. Our main tool is the local bifurcation research of Mf for the level set parametrization fr(ζ) = f(rζ), r > 0.