On Bloch seminorm of finite Blaschke products in the unit disk

Abstract

We prove that, for any finite Blaschke product $w=B(z)$ in the unit disk, the corresponding Riemann surface over the $w$--plane contains a one-sheeted disk of the radius $0.5$. Moreover, it contains a unit one-sheeted disk with a radial slit. We apply this result to obtain a universal sharp lower estimate of the Bloch seminorm for finite Blaschke products.