On Bloch seminorm of finite Blaschke products in the unit disk

TL;DR

This paper establishes a universal lower bound for the Bloch seminorm of finite Blaschke products in the unit disk, revealing geometric properties of their associated Riemann surfaces.

Contribution

It proves the existence of specific geometric structures in the Riemann surface of finite Blaschke products and derives a sharp lower estimate for their Bloch seminorm.

Findings

Riemann surface contains a one-sheeted disk of radius 0.5

Contains a unit one-sheeted disk with a radial slit

Provides a universal sharp lower estimate of the Bloch seminorm

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.