Sharp Bounds of Eigenvalue Ratios for Sturm–Liouville Equations With Dirichlet Boundary Conditions

The aim of this article is to prove the maximization of eigenvalue ratios for the general Sturm–Liouville problem: with Dirichlet boundary conditions, where is a nonnegative continuous potential on and the weight function satisfies that and for some constants and . By finding the explicit solution of the minimization problem of any Dirichlet eigenvalue for the measure differential equation where and are two suitable measures, we solve the infinitely dimensional maximization problem and the solution is given by an elementary function.