Power convexity of the torsion function on horo-convex domains in hyperbolic space

In this paper, we study the torsion problem on bounded, smooth, strictly horo-convex domains in hyperbolic space. We prove that if the diameter of the domain is sufficiently small, then $-\sqrt{u}$ is strictly convex, where $u$ denotes the torsion function. The main ingredients of our proof are the constant rank theorem, a boundary convexity estimate, and a deformation argument for the domain. We also show that the exponent $1\mathrm{/}2$ is optimal, even among strictly horo-convex domains of arbitrarily small diameter.