Super Log-concavity of the First Eigenfunctions for Horo-convex Domains in Hyperbolic Space

In this paper, we prove that the first eigenfunction of the Laplacian for a horo-convex domain $\mathrm{\Omega }\subset {\mathbb{H}}^n$ is super log-concave when $\text{diam}(\mathrm{\Omega })$ is not large. Our result is optimal in the sense that there are counterexamples %are constructed for the cases when $\mathrm{\Omega }$ is not horo-convex or when $\text{diam}(\mathrm{\Omega })$ is large respectively