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<p>This paper focuses on establishing new generalizations of Hilbert-type inequalities on arbitrary time scales. We present three main theorems on weighted integral inequalities involving a nonnegative homogeneous kernel. The proofs rely on several auxiliary lemmas together with the effective application of Hölder’s inequality. By specializing our results to the continuous time scale (<InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> <mo>=</mo> <mi mathvariant="double-struck">R</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{T}=\mathbb{R}$</EquationSource> </InlineEquation>) and the discrete time scale (<InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> <mo>=</mo> <mi mathvariant="double-struck">N</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{T}=\mathbb{N}$</EquationSource> </InlineEquation>), we derive a number of corollaries that unify and extend both classical and recent inequalities. Overall, this work contributes to the theory of integral inequalities by providing a broader framework and new analytical tools within the calculus on time scales.</p>

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Generalizations of Hilbert-type inequalities with a homogeneous kernel on time scales

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月9日

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  • Karim A. Mohamed,
  • Haytham M. Rezk,
  • Ahmed M. Ahmed,
  • Belal A. Glalah

摘要

This paper focuses on establishing new generalizations of Hilbert-type inequalities on arbitrary time scales. We present three main theorems on weighted integral inequalities involving a nonnegative homogeneous kernel. The proofs rely on several auxiliary lemmas together with the effective application of Hölder’s inequality. By specializing our results to the continuous time scale ( T = R $\mathbb{T}=\mathbb{R}$ ) and the discrete time scale ( T = N $\mathbb{T}=\mathbb{N}$ ), we derive a number of corollaries that unify and extend both classical and recent inequalities. Overall, this work contributes to the theory of integral inequalities by providing a broader framework and new analytical tools within the calculus on time scales.