A Unified Approach to Hardy-Type Integral Inequalities for Bivariate Functions on Time Scales

Abstract

This paper establishes several new bivariate Hardy-type dynamic integral inequalities on arbitrary time scales. By utilizing delta integration, properties of rd-continuous functions, and kernel estimates, we provide a unified framework that extends existing continuous two-dimensional inequalities and generalizes their one-dimensional dynamic counterparts. The established inequalities successfully bridge the gap between continuous and discrete analysis, reducing to classical continuous cases when T=R and to their discrete analogues when T=N. These results offer powerful analytical tools for establishing qualitative properties of multi-dimensional dynamic equations.