Power-logconcavity of the Laplacian ground state

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