A new estimate for the constants of an inequality due to Hardy and Littlewood

Antonio Gomes Nunes

Published 2017-01-08Version 1

One of the classical Hardy--Littlewood inequalities for $m$-linear forms on $\ell _{p}$ spaces asserts that \begin{equation*} \left( \sum_{j_{1},...,j_{m}=1}^{\infty }\left\vert T\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq 2^{\frac{m-1}{2}}\left\Vert T\right\Vert \end{equation*}% for all continuous $m$-linear forms $T:\ell _{p}\times \cdots \times \ell _{p}\rightarrow \mathbb{R}$ or $\mathbb{C}$ when $m<p\leq 2m.$ The case $m=2$ recovers an important inequality proved by Hardy and Littlewood in 1934. The search of optimal constants for these kind of inequalities is a fashionable subject, sometimes with concrete and unexpected applications in different fields of Mathematics and Physics, like, for instance, Quantum Information Theory. It was recently proved by Albuquerque \textit{et al.} that $2^{\frac{% m-1}{2}}$ can be improved to $2^{\frac{\left( m-1\right) \left( p-m\right) }{% p}}$. In the present paper we give a step further improving this last estimate to $2^{\frac{\left( p-2\right) \left( m-1\right) \left( p-m\right) }{p^{2}}}$.