On a problem of Polya and Szegő

We give a new proof of a theorem, which is originally due to Gehring and Pommerenke on the triviality of the extrema set $M_f$ of the inner mapping radius $|f'(\zeta)|(1-|\zeta|^2)$ 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 $M_f$ for the level set parametrization $f_r(\zeta)=f(r\zeta)$, $r>0$.