# Sharpness of some Hardy-type inequalities

## TL;DR

This paper reviews Hardy-type inequalities with sharp constants, introduces new inequalities for monotone functions, and explores their implications for Lorentz space quasi-norms, all unified through convexity methods.

## Contribution

It presents a unified convexity approach to Hardy inequalities, introduces new sharp inequalities for monotone functions, and applies these results to Lorentz space quasi-norm relations.

## Key findings

- Established sharp Hardy inequalities with convexity methods
- Derived new two-sided inequalities for monotone functions
- Provided sharp bounds in Lorentz space quasi-norm comparisons

## Abstract

The current status concerning Hardy-type inequalities with sharp constants is presented and described in a unified convexity way. In particular, it is then natural to replace the Lebesgue measure $dx$ with the Haar measure $dx/x.$ There are also derived some new two-sided Hardy-type inequalities for monotone functions, where not only the two constants are sharp but also where the involved function spaces are (more) optimal. As applications, a number of both well-known and new Hardy-type inequalities are pointed out. And, in turn, these results are used to derive some new sharp information concerning sharpness in the relation between different quasi-norms in Lorentz spaces.