  It is well-known that the optimal constant of the bilinear Bohnenblust--Hille
inequality (i.e., Littlewood's $4/3$ inequality) is obtained by interpolating
the bilinear mixed $\left( \ell _{1},\ell_{2}\right) $-Littlewood inequalities.
We remark that this cannot be extended to the $3$-linear case and, in the
opposite direction, we show that the asymptotic growth of the constants of the
$m$-linear Bohnenblust--Hille inequality is the same of the constants of the
mixed $\left( \ell _{\frac{2m+2}{m+2}},\ell _{2}\right) $-Littlewood
inequality. This means that, contrary to what the previous works seem to
suggest, interpolation does not play a crucial role in the search of the exact
asymptotic growth of the constants of the Bohnenblust--Hille inequality. In the
final section we use mixed Littlewood type inequalities to obtain the optimal
cotype constants of certain sequence spaces.
