The eigenvalue gaps and eigenvalue ratios of the Sturm-Liouville systems have been studied in many papers. Recently, Lavine proved an optimal lower estimate of first eigenvalue gaps for Schrodinger operators with convex potentials. His method uses a variational approach with detailed analysis on different integrals. In 1999, (M.J.) Huang adopted his method to study eigenvalue ratios of vibrating strings. He proved an optimal lower estimate of first eigenvalue ratios with nonnegative densities. In this thesis, we want to generalize the above optimal estimate. The work of Ashbaugh and Benguria helps in attaining our objective. They introduced an approach involving a modified Prufer substitution and a comparison theorem to study the upper bounds of Dirichlet eigenvalue ratios for Schrodinger operators with nonnegative potentials. It is interesting to see that the counterpart of their result is also valid. By Liouville substitution and an approximation theorem, the vibrating strings with concave and positive densities can be transformed to a Schrodinger operator with nonpositive potentials. Thus we have the generalization of Huang's result.