On convex domains in R^n and S^n, the first Dirichlet eigenfunction is known to be log concave, a fact that is crucial to estimate the spectral gap, which is the difference between the second and first Dirichlet eigenvalue. It is known that the first Dirichlet eigenfunction is in general not log-concave for convex domains in H^n. I will discuss concavity estimates on horoconvex domains in hyperbolic space - which are domains whose boundaries second fundamental form is greater than 1 -  which yield new spectral-gap bounds in H^n. In doing so, we resolve a conjecture by Nguyen, Stancu and Wei. This is based on joint work with G. Khan and on joint work with S. Saha and G. Khan.