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

Abstract

Abstract 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 ( $\mathbb{T}=\mathbb{R}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> </mml:math> ) and the discrete time scale ( $\mathbb{T}=\mathbb{N}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> </mml:math> ), 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.