Power-logconcavity of the Laplacian ground state

Let $u$ be the first Dirichlet Laplacian eigenfunction of a bounded convex set $\Omega$ in $\mathbb{R}^n$. We strengthen the classical result by Brascamp-Lieb which asserts that $u$ is logconcave in $\Omega$: we prove that, if $u$ is normalized so that its $L^\infty$-norm does not exceed a threshold $\overline{\kappa} (\Omega)<1$ depending explicitly on the diameter of the domain and on its principal frequency, the function $- ( - \log u ) ^{1/2}$ is concave in $\Omega$.