Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$K\subset \mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>⊂</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> be a convex compact set, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Pi _n(K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Π</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the class of polynomials of exact degree n, all of whose zeros lie in K. The Turán type inverse Markov factor is defined by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$M_n(K)=\inf _{P\in \Pi _n(K)} \left( \Vert P'\Vert _{C(K)}/\Vert P\Vert _{C(K)}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo movablelimits="true">inf</mo> <mrow> <mi>P</mi> <mo>∈</mo> <msub> <mi mathvariant="normal">Π</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mfenced close=")" open="("> <mrow> <mo stretchy="false">‖</mo> </mrow> <msup> <mi>P</mi> <mo>′</mo> </msup> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>P</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. A combination of two well-known results due to N. Levenberg and E.A. Poletsky, Reverse Markov inequality, Ann. Acad. Sci. Fenn.Math. 27, 173–182 (2002) and Sz.Gy Révész, Turán type reverse Markov inequalities for compact convex sets. J. Approx. Theory 141, 162–173 (2006) provides the lower bound <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M_n(K)\ge c\left( wn/d^2+\sqrt{n}/d\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <mi>c</mi> <mfenced close=")" open="("> <mi>w</mi> <mi>n</mi> <mo stretchy="false">/</mo> <msup> <mi>d</mi> <mn>2</mn> </msup> <mo>+</mo> <msqrt> <mi>n</mi> </msqrt> <mo stretchy="false">/</mo> <mi>d</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$c:=0.00015$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>:</mo> <mo>=</mo> <mn>0.00015</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$d>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is the diameter of K and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$w\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is the minimal width (the smallest distance between two parallel lines between which K lies). We prove that this bound is essentially sharp, namely, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$M_n(K)\le 28\left( wn/d^2+\sqrt{n}/d\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mn>28</mn> <mfenced close=")" open="("> <mi>w</mi> <mi>n</mi> <mo stretchy="false">/</mo> <msup> <mi>d</mi> <mn>2</mn> </msup> <mo>+</mo> <msqrt> <mi>n</mi> </msqrt> <mo stretchy="false">/</mo> <mi>d</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> for all n, w, d.

On the precise form of the inverse Markov factor for convex sets

期刊论文
发表日期：

2025年11月14日

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Mikhail Komarov

摘要

Let $K\subset \mathbb {C}$ $K \subset C$ be a convex compact set, and let $\Pi _n(K)$ $Π_{n} (K)$ be the class of polynomials of exact degree n, all of whose zeros lie in K. The Turán type inverse Markov factor is defined by $M_n(K)=\inf _{P\in \Pi _n(K)} \left( \Vert P'\Vert _{C(K)}/\Vert P\Vert _{C(K)}\right) $ $M_{n} (K) = {inf}_{P \in Π_{n} (K)} (‖ P^{'} ‖_{C (K)} / {‖ P ‖}_{C (K)})$ . A combination of two well-known results due to N. Levenberg and E.A. Poletsky, Reverse Markov inequality, Ann. Acad. Sci. Fenn.Math. 27, 173–182 (2002) and Sz.Gy Révész, Turán type reverse Markov inequalities for compact convex sets. J. Approx. Theory 141, 162–173 (2006) provides the lower bound $M_n(K)\ge c\left( wn/d^2+\sqrt{n}/d\right) $ $M_{n} (K) \geq c (w n / d^{2} + \sqrt{n} / d)$ , $c:=0.00015$ $c : = 0.00015$ , where $d>0$ $d > 0$ is the diameter of K and $w\ge 0$ $w \geq 0$ is the minimal width (the smallest distance between two parallel lines between which K lies). We prove that this bound is essentially sharp, namely, $M_n(K)\le 28\left( wn/d^2+\sqrt{n}/d\right) $ $M_{n} (K) \leq 28 (w n / d^{2} + \sqrt{n} / d)$ for all n, w, d.