Levin-Cochran-Lee inequalities and best constants on homogeneous groups

TL;DR

This paper establishes Levin-Cochran-Lee inequalities on homogeneous groups using a direct proof method, deriving sharp constants and new inequalities for specific parameter cases.

Contribution

It introduces a direct proof approach for Levin-Cochran-Lee inequalities on homogeneous groups and finds sharp constants for the case p=q.

Findings

Proved Levin-Cochran-Lee inequalities on homogeneous groups.

Derived sharp inequalities with power weights for p=q.

Established new inequalities beyond existing results.

Abstract

In this paper, we apply a direct method instead of a limit approach, for proving the Levin-Cochran-Lee inequalities. First, we state and prove Levin-Cochran-Lee type inequalities on a homogeneous group $\mathbb{G}$ with parameters $0<p\leq q<\infty$. Furthermore, for the case $p=q$, we prove the sharp inequalities with power weights and derive some other new inequalities.