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

TL;DR

This paper establishes a precise inequality for the measure of sublevel sets of finite Blaschke products, revealing conditions for equality and extending understanding of their geometric properties.

Contribution

It provides a sharp estimate for the area of sublevel sets of finite Blaschke products, including conditions for equality, which was previously unknown.

Findings

The Lebesgue measure of sublevel sets satisfies a sharp inequality.

Equality holds only when all zeros are at the origin.

The inequality is tight and characterizes extremal cases.

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 $t$.