Scholarlink

<p>We study a special case of the Markov-Nikol’skii inequality for the class of algebraic polynomials with complex coefficients that do not vanish in the disc. This inequality estimates the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^q\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>-mean of the derivative of order <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(k\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> </InlineEquation> of a polynomial on the interval <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\([-1,1]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> from above by the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(L^0\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation>-mean (geometric mean) of the polynomial itself on the same interval. We obtain the exact constant in this inequality for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(q\ge2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and arbitrary <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(k\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> </InlineEquation> and characterize all extremal polynomials. </p><p>Additionally, for the class of polynomials with zeros in the closed disk, we study the extremal case of Turán's inequality when the <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(L^r\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation>-mean on <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\([-1,1] \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> of the derivative of order <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\( k=n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> (the degree of the polynomial) is estimated from below by the <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(L^q\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>-mean of the polynomial itself on the interval. We obtain the exact constant in this inequality for all <InlineEquation ID="IEq17"> <EquationSource Format="TEX">\(0 \le r, q \le \infty\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>r</mi> <mo>,</mo> <mi>q</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and describe the set of extremal polynomials.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Inequality for \(L^q\)-means of the derivative and \(L^0\)-means of polynomials with zeros outside the disk

  • 期刊论文

  • 发表日期：

    2026年8月4日

查看全文
  • A. E. Rokina

摘要

We study a special case of the Markov-Nikol’skii inequality for the class of algebraic polynomials with complex coefficients that do not vanish in the disc. This inequality estimates the \(L^q\) L q -mean of the derivative of order \(k\) k of a polynomial on the interval \([-1,1]\) [ - 1 , 1 ] from above by the \(L^0\) L 0 -mean (geometric mean) of the polynomial itself on the same interval. We obtain the exact constant in this inequality for \(q\ge2\) q ≥ 2 and arbitrary \(k\) k and characterize all extremal polynomials.

Additionally, for the class of polynomials with zeros in the closed disk, we study the extremal case of Turán's inequality when the \(L^r\) L r -mean on \([-1,1] \) [ - 1 , 1 ] of the derivative of order \( k=n\) k = n (the degree of the polynomial) is estimated from below by the \(L^q\) L q -mean of the polynomial itself on the interval. We obtain the exact constant in this inequality for all \(0 \le r, q \le \infty\) 0 ≤ r , q ≤ ∞ and describe the set of extremal polynomials.