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

On the mixed $\left( \ell _{1},\ell _{2}\right)$-Littlewood inequality for real scalars and applications