An optimal inequality between nonlinear eigenvalues in convex sets

Let $λ_p(Ω)$ be the first Dirichlet eigenvalue of the $p$-Laplacian in a bounded convex open set $Ω\subset\mathbb R^N$, $N\ge2$. We prove that $p\mapstoλ_p(Ω)^{1/p}/π_p$ is strictly decreasing, where $π_p^p$ is the first eigenvalue on the unit interval. Consequently, for $1<\infty$, the sharp upper bound for $λ_p(Ω)^{1/p}/λ_q(Ω)^{1/q}$ is $π_p/π_q$. 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.