Abstract <p> Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(B\)</EquationSource> </InlineEquation> be a finite Blaschke product of degree <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n\geq 2\)</EquationSource> </InlineEquation>. We discuss the behavior of the area of the set <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(E(B,t)=\{z\colon |B(z)|\leq t\}\)</EquationSource> </InlineEquation> as the parameter <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(t\)</EquationSource> </InlineEquation> changes, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(0<t<1\)</EquationSource> </InlineEquation>, and also a condition on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(t\)</EquationSource> </InlineEquation> under which the connected component of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(E(B,t)\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(B(0)=0\)</EquationSource> </InlineEquation>, containing the origin contains at least one critical point of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(B\)</EquationSource> </InlineEquation>. For real products <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(B\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(B(0)=0\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(B'(0)\not= 0\)</EquationSource> </InlineEquation>, we establish sharp lower bounds for the moduli of some critical points in terms of the moduli of the corresponding critical values and the quantity <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(|B'(0)|\)</EquationSource> </InlineEquation>. Cases are considered separately in which the critical points are located on the same radius of the disk <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(|z|<1\)</EquationSource> </InlineEquation> and in which they lie on the same diameter of this disk on different sides of the origin. These estimates do not depend on the degree of the product <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(B\)</EquationSource> </InlineEquation>. </p>

On Lemniscates and Critical Points of Finite Blaschke Products

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发表日期：

2025年10月1日

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V. N. Dubinin

摘要

Abstract

Let \(B\) be a finite Blaschke product of degree \(n\geq 2\) . We discuss the behavior of the area of the set \(E(B,t)=\{z\colon |B(z)|\leq t\}\) as the parameter \(t\) changes, \(0<t<1\) , and also a condition on \(t\) under which the connected component of \(E(B,t)\) , \(B(0)=0\) , containing the origin contains at least one critical point of \(B\) . For real products \(B\) , \(B(0)=0\) , \(B'(0)\not= 0\) , we establish sharp lower bounds for the moduli of some critical points in terms of the moduli of the corresponding critical values and the quantity \(|B'(0)|\) . Cases are considered separately in which the critical points are located on the same radius of the disk \(|z|<1\) and in which they lie on the same diameter of this disk on different sides of the origin. These estimates do not depend on the degree of the product \(B\) .