Sharp upper bound for the first non-zero Neumann eigenvalue for bounded domains in rank-1 symmetric spaces

In this paper, we prove that for a bounded domain $\Omega$ in a rank-$1$ symmetric space, the first non-zero Neumann eigenvalue $\mu _{1}(\Omega )\leq \mu _{1}(B(r_{1}))$ where $B(r_{1})$ denotes the geodesic ball of radius $r_{1}$ such that

and equality holds iff $\Omega =B(r_{1})$. This result generalises the works of Szego, Weinberger and Ashbaugh-Benguria for bounded domains in the spaces of constant curvature.