On the mixed left( ell ₁,ell ₂right)-Littlewood inequality for real scalars and applications

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.