I'm looking for an upper bound on the first nonzero eigenvalue of the LaplaceBeltrami operator on compact manifolds of dimension greater than four that have constant negative curvature. In particular, I would like to know whether or not there is an upper bound that is less than $\frac{1}{2}n^2  2$, where $n$ is the dimension, for the case when the sectional curvature is $1$. I can find quite a few papers on lower bounds, but the few I could find on upper bounds required information that is hard to obtain. Any help would be greatly appreciated.
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Did you look in Isaac Chavel's book 'Eigenvalues in Riemannian geometry'? It contains various bounds for eigenvalues although at a quick glance I didn't see anything for your particular setting. 

