On the upper bounds for the constants of the Hardy-Littlewood inequality
  The best known upper estimates for the constants of the Hardy--Littlewood inequality for $m$-linear forms on $\ell_{p}$ spaces are of the form $\left(\sqrt{2}\right) ^{m-1}.$ We present better estimates which depend on $p$ and $m$. An interesting consequence is that if $p\geq m^{2}$ then the constants have a subpolynomial growth as $m$ tends to infinity. 
Gustavo Araujo, Daniel Pellegrino, Diogo D.P. Silva e Silva
math.FA