Optimal constants for a mixed Littlewood type inequality

For $p\in [2,\mathrm{\infty }]$ a mixed Littlewood-type inequality asserts that there is a constant $C_{(m),p}\ge 1$ such that $${\left(\sum _{i_1=1}^{\mathrm{\infty }}{(\sum _{i_2,\mathrm{.}\mathrm{.}\mathrm{.},i_m=1}^{\mathrm{\infty }}\mathrm{\mid }T(e_{i_1},\mathrm{.}\mathrm{.}\mathrm{.},e_{i_m}){\mathrm{\mid }}^2)}^{\frac{1}{2}\frac{p}{p-1}}\right)}^{\frac{p-1}{p}}\le C_{(m),p}\mathrm{\parallel }T\mathrm{\parallel }$$ for all continuous real-valued $m$-linear forms on ${\mathrm{\ell }}_p\times c_0\times \cdots \times c_0$ (when $p=\mathrm{\infty }$, ${\mathrm{\ell }}_p$ is replaced by $c_0)$. We prove that for $p>2.18006$ the optimal constants $C_{(m),p}$ are $\left( 2^{\frac{1}{2}-\frac{1}{p}}\right) ^{m-1}.$When$p=\infty,$ we recover the best constants of the mixed $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$-Littlewood inequality.