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In this note, among other results, we find the optimal constants of thegeneralized Bohnenblust--Hille inequality for $m$-linear forms over$\mathbb{R}$ and with multiple exponents $\left( 1,2,...,2\right) $, sometimescalled mixed $\left( \ell _{1},\ell _{2}\right) $-Littlewood inequality. Weshow that these optimal constants are precisely $\left( \sqrt{2}\right) ^{m-1}$and this is somewhat surprising since a series of recent papers have shown thatsimilar constants have a sublinear growth. This result answers a questionraised by Albuquerque \textit{et al.} in a paper published in 2014 in the\textit{Journal of Functional Analysis}.

The optimal constants of the mixed $\left( \ell_{1},\ell _{2}\right) $-Littlewood inequality