  It was recently proved that for $p>2m^{3}-4m^{2}+2m$ the constants of the
Hardy--Littlewood inequality for $m$-linear forms on $\ell_{p}$-spaces are less
than or equal to the best known estimates of respective constants of the
Bohnenblust--Hille inequality. In this note we obtain upper bounds for opposite
side, i.e., the constants when $2m\leq p\leq2m^{3}-4m^{2}+2m.$ For these values
of $p$ our result improves previous estimates from 2014 of Araujo \textit{et
al}. for all $m\geq3.$
