Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation> be the unit disk in the complex plane. Among other results, we prove the following curious result for a finite Blaschke product: <Equation ID="Equ8"> <EquationSource Format="TEX">$ B(z)=e ^{is}\prod _{k=1}^d \frac{z-a_k}{1-z \overline{a_k}}. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mi mathvariant="italic">is</mi> </mrow> </msup> <munderover> <mo>∏</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>d</mi> </munderover> <mfrac> <mrow> <mi>z</mi> <mo>-</mo> <msub> <mi>a</mi> <mi>k</mi> </msub> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mi>z</mi> <mover> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>¯</mo> </mover> </mrow> </mfrac> <mo>.</mo> </mrow> </math></EquationSource> </Equation>The Lebesgue measure of the sublevel set of B satisfies the following sharp inequality for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t \in [0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>: <Equation ID="Equ9"> <EquationSource Format="TEX">$ |\{z\in \mathbb {D}:|B(z)|<t\}|\le \pi t^{2/d}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">{</mo> <mi>z</mi> </mrow> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">D</mi> <mo>:</mo> <mo stretchy="false">|</mo> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mo><</mo> <mi>t</mi> <mo stretchy="false">}</mo> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>π</mi> <msup> <mi>t</mi> <mrow> <mn>2</mn> <mo stretchy="false">/</mo> <mi>d</mi> </mrow> </msup> <mo>,</mo> </mrow> </math></EquationSource> </Equation>with equality at a single point <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$t\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a_k=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for every k. In that case the equality is attained for every t.

A Sharp Estimate of Area for Sublevel Set of Blaschke Products

期刊论文
发表日期：

2026年1月30日

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David Kalaj

摘要

Let $\mathbb {D}$ $D$ 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}}. $ $B (z) = e^{is} \prod_{k = 1}^{d} \frac{z - a_{k}}{1 - z \bar{a_{k}}} .$ The Lebesgue measure of the sublevel set of B satisfies the following sharp inequality for $t \in [0,1]$ $t \in [0, 1]$ : $ |\{z\in \mathbb {D}:|B(z)|<t\}|\le \pi t^{2/d}, $ $| {z \in D : | B (z) | < t} | \leq π t^{2 / d},$ with equality at a single point $t\in (0,1)$ $t \in (0, 1)$ if and only if $a_k=0$ $a_{k} = 0$ for every k. In that case the equality is attained for every t.