  The Hardy--Littlewood inequalities for $m$-linear forms have their origin
with the seminal paper of Hardy and Littlewood (Q.J. Math, 1934). Nowadays it
has been extensively investigated and many authors are looking for the optimal
estimates of the constants involved. For $m<p\leq2m$ it asserts that there is a
constant $D_{m,p}^{\mathbb{K}}\geq1$ such that \[ \left(
\sum_{j_{1},\cdots,j_{m}=1}^{n}\left\vert
T(e_{j_{1}},\cdots,e_{j_{m}})\right\vert ^{\frac{p}{p-m}}\right)
^{\frac{p-m}{p}}\leq D_{m,p}^{\mathbb{K}}\left\Vert T\right\Vert , \] for all
$m$--linear forms
$T:\ell_{p}^{n}\times\cdots\times\ell_{p}^{n}\rightarrow\mathbb{K}=\mathbb{R}$
or $\mathbb{C}$ and all positive integers $n$. Using a Regularity Principle
recently proved by Pellegrino, Santos, Serrano and Teixeira, we present a
straightforward proof of the Hardy--Littewood inequality and show that:
  (1) If $m<p_{1}<p_{2}\leq2m$ then $D_{m,p_{1}}^{\mathbb{K}}\leq
D_{m,p_{2}}^{\mathbb{K}}$;
  (2) $D_{m,p}^{\mathbb{K}}\leq D_{m-1,p}^{\mathbb{K}}$ whenever $m<p\leq
2\left( m-1\right) $ for all $m\geq3$.
