  In this paper we obtain the sharp estimates for the mixed $\left(
\ell_{1},\ell _{2}\right) $-Littlewood inequality for real scalars with
exponents $\left(2,1,2,2....,2\right) .$ These results are applied to find
sharp estimates for the constants of a family of $3$-linear Bohnenblust--Hille
inequalities with multiple exponents.
