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<p>In this paper, we study the growth of the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1055_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\(m-th\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>-</mo> <mi>t</mi> <mi>h</mi> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1055_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(m\ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) derivatives of an arbitrary algebraic polynomial in weighted Lebesgue spaces over the whole complex plane. We first study the growth of the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1055_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\(m-th\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>-</mo> <mi>t</mi> <mi>h</mi> </mrow> </math></EquationSource> </InlineEquation> derivatives of an arbitrary algebraic polynomial over unbounded regions of the complex plane, and then we obtain estimates for the growth of the <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1055_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\(m-th\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>-</mo> <mi>t</mi> <mi>h</mi> </mrow> </math></EquationSource> </InlineEquation> derivatives of this polynomial over the closure of the given region. Combining both estimates, we find estimates for the growth of the <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1055_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\(m-th\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>-</mo> <mi>t</mi> <mi>h</mi> </mrow> </math></EquationSource> </InlineEquation> derivatives of an arbitrary algebraic polynomial over the whole complex plane.</p>

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On the behavior of \(m-th\) derivatives of polynomials in bounded and unbounded regions without zero angles in weighted Lebesgue spaces

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  • 发表日期：

    2025年4月18日

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  • F. G. Abdullayev,
  • M. Imashkyzy

摘要

In this paper, we study the growth of the \(m-th\) m - t h ( \(m\ge 1\) m ≥ 1 ) derivatives of an arbitrary algebraic polynomial in weighted Lebesgue spaces over the whole complex plane. We first study the growth of the \(m-th\) m - t h derivatives of an arbitrary algebraic polynomial over unbounded regions of the complex plane, and then we obtain estimates for the growth of the \(m-th\) m - t h derivatives of this polynomial over the closure of the given region. Combining both estimates, we find estimates for the growth of the \(m-th\) m - t h derivatives of an arbitrary algebraic polynomial over the whole complex plane.