A refinement of a Hardy type inequality for negative exponents, and sharp applications to Muckenhoupt weights on R

TL;DR

This paper establishes a sharp, generalized Hardy type inequality for negative exponents and applies it to Muckenhoupt weights on the real line, refining previous results in the field.

Contribution

It introduces a refined, sharp integral inequality for negative exponents and provides improved applications to Muckenhoupt weights on R.

Findings

Proved a sharp generalized Hardy inequality for negative exponents

Provided improved applications to Muckenhoupt weights on R

Refined previous results in the literature

Abstract

We prove a sharp integral inequality that generalizes the well known Hardy type integral inequality for negative exponents. We also give sharp applications in two directions for Muckenhoupt weights on R. This work refines the results that appear in [9].