Log-Concavity of the First Eigenfunction on Horoconvex Domains in the Hyperbolic Plane

Let $\Omega\subset\mathbb H^2$ be a bounded smooth horoconvex domain and let $\psi_1>0$ be its first Dirichlet eigenfunction. We prove that $$\operatorname{Hess}_{\mathbb H^2}(-\log\psi_1)>0$$ throughout $\Omega$, with no restriction on the diameter or the first eigenvalue. The proof is by contradiction. A degenerate Hessian would yield a shifted translation Killing derivative with a singular interior zero. Then the boundary-zero theorem of Grossi and Provenzano shows that the shifted Killing derivative has exactly two zeros on the boundary. A nodal-domain argument on the surface rules this out.