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<p>In this study, we use the properties of convex functions, Jensen’s inequality, Holder’s inequality, and the chain rule to establish a new class of dynamic Hardy-type inequalities on time scales using delta calculus. The results include integral inequalities due to Sulaiman, where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2024_7554_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {T=R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">T</mi> <mo>=</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, as specific cases of our results. We can also find further inequalities on different time scales, such as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2024_7554_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {T=Z}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">T</mi> <mo>=</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation>, which are, as far as the authors know, recent results.</p>

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DYNAMIC HARDY-TYPE INEQUALITIES WITH CONVEX AND NON-CONVEX FUNCTIONS ON TIME SCALES

  • 期刊论文

  • 发表日期：

    2025年1月30日

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  • Haytham M. Rezk,
  • Hassan M. El-Owaidy,
  • Wafy M. Hasan,
  • Ahmed A. El-Deeb

摘要

In this study, we use the properties of convex functions, Jensen’s inequality, Holder’s inequality, and the chain rule to establish a new class of dynamic Hardy-type inequalities on time scales using delta calculus. The results include integral inequalities due to Sulaiman, where \(\mathbb {T=R}\) T = R , as specific cases of our results. We can also find further inequalities on different time scales, such as \(\mathbb {T=Z}\) T = Z , which are, as far as the authors know, recent results.