  The Hardy--Littlewood inequalities for multilinear forms on sequence spaces
state that for all positive integers $m,n\geq2$ and all $m$-linear forms
$T:\ell_{p_{1}}^{n}\times\cdots\times\ell_{p_{m}}^{n}\rightarrow\mathbb{K}$
($\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$) there are constants $C_{m}\geq1$ (not
depending on $n$) such that \[ \left( \sum_{j_{1},\ldots,j_{m}=1}^{n}\left\vert
T(e_{j_{1}},\ldots,e_{j_{m}})\right\vert ^{\rho}\right) ^{\frac{1}{\rho}}\leq
C_{m}\sup_{\left\Vert x_{1}\right\Vert ,\dots,\left\Vert x_{m}\right\Vert \leq
1}\left\vert T(x_{1},\dots,x_{m})\right\vert, \] where
$\rho=\frac{2m}{m+1-2\left( \frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}\right) }$ if
$0\leq\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}\leq\frac{1}{2}$ or
$\rho=\frac{1}{1-\left( \frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}\right)}$ if
$\frac{1}{2}\leq\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}<1$. Good estimates for
the Hardy-Littlewood constants are, in general, associated to applications in
Mathematics and even in Physics, but the exact behavior of these constants is
still unknown. In this note we give some new contributions to the behavior of
the constants in the case
$\frac{1}{2}\leq\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}<1$. As a consequence of
our main result, we present a generalization and a simplified proof of a result
due to Aron et al. on certain Hardy--Littlewood type inequalities.
