An optimal inequality between nonlinear eigenvalues in convex sets

Let ${\lambda }_p(\mathrm{\Omega })$ be the first Dirichlet eigenvalue of the $p$-Laplacian in a bounded convex open set $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\ge 2$. We prove that $p\mapsto {\lambda }_p(\mathrm{\Omega }{)}^{1\mathrm{/}p}\mathrm{/}{\pi }_p$ is strictly decreasing, where ${\pi }_p^p$ is the first eigenvalue on the unit interval. Consequently, for $1<q<p<\mathrm{\infty }$, the sharp upper bound for ${\lambda }_p(\mathrm{\Omega }{)}^{1\mathrm{/}p}\mathrm{/}{\lambda }_q(\mathrm{\Omega }{)}^{1\mathrm{/}q}$ is ${\pi }_p\mathrm{/}{\pi }_q$. This bound is not attained and is approached by slabs, resolving the conjecture on the supremum formulated in the plane by Parini (J. Convex Anal. 2017). The proof combines log-concavity of the first eigenfunction with a vector field constructed from the one-dimensional eigenfunctions. The endpoint limits recover the Hersch–Protter and reverse Cheeger (or Buser) inequalities.