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 ${\mathrm{\ell }}_p$ spaces are of the form ${\left(\sqrt{2}\right)}^{m-1}\mathrm{.}$ We present better estimates which depend on $p$ and $m$. An interesting consequence is that if $p\ge m^2$ then the constants have a subpolynomial growth as $m$ tends to infinity.