  Buser's inequality gives an upper bound on the first non-zero eigenvalue of
the Laplacian of a closed manifold M in terms of the Cheeger constant h(M).
Agol later gave a quantitative improvement of Buser's inequality. Agol's result
is less transparent since it is given implicitly by a set of equations, one of
which is a differential equation Agol could not solve except when M is
three-dimensional. We show that a substitution transforms Agol's differential
equation into the Riemann differential equation. Then, we give a proof of
Agol's result and also generalize it using Sturm-Liouville theory. Under the
same assumptions on M, we are able to give upper bounds on the higher
eigenvalues of M, \lambda_k(M), in terms of the eigenvalues of a
Sturm-Liouville problem which depends on h(M). We then compare the Weyl
asymptotic of \lambda_k(M) given by the works of Cheng, Gromov, and
B\'erard-Besson-Gallot to the asymptotics of our Sturm-Liouville problems given
by Atkinson-Mingarelli.
