A nontrivial closed formula for the lower bounds for the constants of the real Hardy-Littlewood inequalities

TL;DR

This paper derives a simple, closed-form formula to improve lower bounds for the constants in the Hardy-Littlewood inequalities for m-linear forms on bcl_p spaces, covering key cases with enhanced estimates.

Contribution

It introduces a new, straightforward method using Clarkson's inequalities to obtain improved lower bounds and a unified closed-form formula for these constants.

Findings

Enhanced lower bounds for Hardy-Littlewood inequality constants

Unified closed-form formula for p > 2m and p = 2m cases

Simpler approach compared to previous methods

Abstract

This note has a twofold purpose. To improve the best known lower estimates of the Hardy-Littlewood inequality for $m$-linear forms in $\ell_{p}$ spaces and to provide a closed formula encompassing the cases $p>2m$ and $% p=2m.$ Our approach is quite simpler than the previous and uses the best known constants of the Clarkson's inequalities.