The optimal constant in Hardy-type inequalities

TL;DR

This paper introduces new variational formulas and approximation methods to better estimate the optimal constants in Hardy-type inequalities, improving known bounds and illustrating their significance through examples.

Contribution

It provides novel variational formulas and approximation procedures that significantly enhance the estimation of optimal constants in Hardy-type inequalities.

Findings

Improved basic estimates for Hardy-type inequalities

Demonstrated the relevance of the sharp factor on finite intervals

Re-examined classical sharp models with new insights

Abstract

To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined.