  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.
