  The Hardy--Littlewood inequality for $m$-linear forms on $\ell _{p}$ spaces
and $m<p\leq 2m$ 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 $\mathbb{C}.$ The case $m=2$ recovers a classical inequality
proved by Hardy and Littlewood in 1934. As a consequence of the results of the
present paper we show that the same inequality is valid with
$2^{\frac{m-1}{2}}$ replaced by $2^{\frac{\left( m-1\right) \left( p-m\right)
}{p}}$. In particular, for $m<p\leq m+1$ the optimal constants of the above
inequality are uniformly bounded by $2.$
