Exploring Generalized Hardy-Type Inequalities via Nabla Calculus on Time Scales

Abstract

In this research, we aim to explore generalizations of Hardy-type inequalities using nabla Hölder’s inequality, nabla Jensen’s inequality, chain rule on nabla calculus and leveraging the properties of convex and submultiplicative functions. Nabla calculus on time scales provides a unified framework that unifies continuous and discrete calculus, making it a powerful tool for studying various mathematical problems on time scales. By utilizing this approach, we seek to extend Hardy-type inequalities beyond their classical continuous or discrete settings to a more general time scale domain. As specific instances of our discoveries, we have the integral inequalities previously established in the existing literature.

Keywords

• Hardy’s inequality
• Hölder’s inequality
• Jensen’s inequality
• convex functions
• submultiplicative functions
• nabla calculus