Improvements and generalizations of two Hardy type inequalities and their applications to the Rellich type inequalities

TL;DR

This paper presents improved and generalized Hardy inequalities, deriving new inequalities with optimal constants and applying these to enhance Rellich inequalities, advancing the theoretical framework in mathematical analysis.

Contribution

The paper introduces novel improvements and generalizations of Hardy and Rellich inequalities, including optimal constants and broader applicability.

Findings

Derived Hardy inequalities with best constants

Enhanced Rellich inequalities using improved Hardy inequalities

Unified approach based on divergence theorem

Abstract

We give improvements and generalizations of both the classical Hardy inequality and the geometric Hardy inequality based on the divergence theorem. Especially, our improved Hardy type inequality derives both two Hardy type inequalities with best constants. Besides, we improve two Rellich type inequalities by using the improved Hardy type inequality.