  Let $\mathbb{M}$ denote a complete, simply connected Riemannian manifold with
sectional curvature $K_{\mathbb{M}} \leq k$ and Ricci curvature
$\text{Ric}_{\mathbb{M}} \geq (n-1)K$, where $k,K \in \mathbb{R}$. Then for a
bounded domain $\Omega \subset\mathbb{M}$ with smooth boundary, we prove that
the first nonzero Neumann eigenvalue $\mu_{1}(\Omega) \leq \mathcal{C}
\mu_{1}(B_{k}(R))$. Here $B_{k}(R)$ is a geodesic ball of radius $R > 0$ in the
simply connected space form $\mathbb{M}_{k}$ such that vol$(\Omega)$ =
vol$(B_{k}(R))$, and $\mathcal{C}$ is a constant which depends on the volume,
diameter of $\Omega$ and the dimension of $\mathbb{M}$.
