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<p>In this paper, we explore novel Wirtinger-type inequalities for the Hilfer fractional derivatives. By formulating suitable integral identities and employing Hölder’s inequality, we propose a class of generalized Hilfer fractional Wirtinger-type inequalities for the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3372_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>s</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$L_{s}$</EquationSource> </InlineEquation> space with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3372_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>s</mi> <mo>&gt;</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$s&gt;1$</EquationSource> </InlineEquation>. Several special cases for the Riemann-Liouville and Caputo fractional integral inequalities are also presented. The validity of the findings is illustrated through examples and graphical representations. Moreover, applications are given in the context of inequalities involving the arithmetic mean and geometric mean.</p>

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Hilfer fractional derivatives and their role in Wirtinger-type inequalities with applications

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月30日

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  • Samraiz Muhammad,
  • Yussouf Areej,
  • Naheed Saima,
  • Etemad Sina,
  • Tariboon Jessada

摘要

In this paper, we explore novel Wirtinger-type inequalities for the Hilfer fractional derivatives. By formulating suitable integral identities and employing Hölder’s inequality, we propose a class of generalized Hilfer fractional Wirtinger-type inequalities for the L s $L_{s}$ space with s > 1 $s>1$ . Several special cases for the Riemann-Liouville and Caputo fractional integral inequalities are also presented. The validity of the findings is illustrated through examples and graphical representations. Moreover, applications are given in the context of inequalities involving the arithmetic mean and geometric mean.