On the mixed $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$-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 $(2,1,2,\mathrm{2....},2)\mathrm{.}$ These results are applied to find sharp estimates for the constants of a family of $3$-linear Bohnenblust–Hille inequalities with multiple exponents.