On Innovative Conditions of Diamond-∝ Inequalities and Their Applications

Abstract

This work employs the techniques to present novel structures to diamond-proportional to Hardy-type inequalities with boundedness. To this end, the properties of sub-multiplicative convex functions, Hölder's inequality, Jensen's inequality, and the chain rule are utilized. Additionally, the findings of this study are integrated with those of time-scale calculus and extended. The results sometimes yield constant-valued counterparts to some inequalities identified in the existing literature.