A sharp estimate of area for sublevel-set of Blaschke products

Abstract

Let $\mathbb{D}$ be the unit disk in the complex plane. Among other results, we prove the following curious result for a finite Blaschke product: $$B(z)=e ^{is}\prod_{k=1}^d \frac{z-a_k}{1-z \overline{a_k}}.$$ The Lebesgue measure of the sublevel set of $B$ satisfies the following sharp inequality for $t \in [0,1]$: $$|\{z\in \mathbb{D}:|B(z)|<t\}|\le \pi t^{2/d},$$ with equality at a single point $t\in(0,1)$ if and only if $a_k=0$ for every $k$. In that case the equality is attained for every $t$.