Novel dynamic inequalities of Hilbert-Pachpatte-type for a class of non-homogeneous kernels on time scales

In this paper, we establish novel dynamic Hilbert–Pachpatte–type inequalities on a time scale T involving a class of non-homogeneous kernels k(s,t) = (λ (s) + ρ (t))η , η > 0 , where λ and ρ are positive functions on T . Our approach combines properties of the Gamma function with Jensen’s and Hölder’s inequalities, together with the time-scale version of Fubini’s theorem. In the special cases T=N and T=R , we recover the classical discrete and continuous inequalities of Batbold et al. in (Appl. Math. Comput. 343:167–182, 2019). Moreover, in the quantum case T=q^N₀ with $q>1$ , the results obtained here are entirely new. The applicability of our results is illustrated by several examples and remarks.