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 $\mathbb{T}$ involving a class of non-homogeneous kernels $k(s,t) = (\lambda (s) + \rho (t))^{\eta}$ , $\eta > 0$ , where λ and ρ are positive functions on $\mathbb{T}$ . 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 $\mathbb{T}=\mathbb{N}$ and $\mathbb{T}=\mathbb{R}$ , 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 $\mathbb{T}=q^{\mathbb{N}_{0}}$ with $q>1$ , the results obtained here are entirely new. The applicability of our results is illustrated by several examples and remarks.