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Buser's inequality gives an upper bound on the first non-zero eigenvalue ofthe 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 resultis less transparent since it is given implicitly by a set of equations, one ofwhich is a differential equation Agol could not solve except when M isthree-dimensional. We show that a substitution transforms Agol's differentialequation into the Riemann differential equation. Then, we give a proof ofAgol's result and also generalize it using Sturm-Liouville theory. Under thesame assumptions on M, we are able to give upper bounds on the highereigenvalues of M, \lambda_k(M), in terms of the eigenvalues of aSturm-Liouville problem which depends on h(M). We then compare the Weylasymptotic of \lambda_k(M) given by the works of Cheng, Gromov, andB\'erard-Besson-Gallot to the asymptotics of our Sturm-Liouville problems givenby Atkinson-Mingarelli.

Sturm-Liouville Estimates for the Spectrum and Cheeger Constant