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Abstract <p>Recently, Rather et al. [<CitationRef CitationID="CR1">1</CitationRef>] introduced an operator <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7223_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(N_{m}\)</EquationSource> <!--ContMath2570007Rather-m1--> </InlineEquation> and established certain sharp Bernstein-type polynomial inequalities. In this paper, we show that the operator <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7223_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(N_{m}\)</EquationSource> <!--ContMath2570007Rather-m2--> </InlineEquation> preserves Zygmund-type polynomial inequalities and thereby extends the results of Rather et al. [<CitationRef CitationID="CR1">1</CitationRef>] to integral norm. Our results not only contain several well known results as special cases but also yield certain new interesting results as special cases.</p>

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Integral Mean Estimates for Operators Preserving Zygmund-Type Inequalities

  • 期刊论文

  • 发表日期：

    2025年6月27日

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  • N. A. Rather,
  • L. Ali,
  • T. Bhat

摘要

Abstract

Recently, Rather et al. [1] introduced an operator \(N_{m}\) and established certain sharp Bernstein-type polynomial inequalities. In this paper, we show that the operator \(N_{m}\) preserves Zygmund-type polynomial inequalities and thereby extends the results of Rather et al. [1] to integral norm. Our results not only contain several well known results as special cases but also yield certain new interesting results as special cases.