Sharp estimates for the eigenvalues of some differential equations

We present optimal upper and lower bounds for the eigenvalues of the differential equations y " ? q(x)y + lambda rho(x)y = 0 and (q(x)y')' + lambda rho(x)y = 0 on a finite interval with Dirichlet boundary conditions when the coefficient functions q(x) and rho(x) are nonnegative and are subjected to some kind of additional constraints. One of the basic ideas used in our work consists in reducing the problem of maximizing lambda(q, rho) to an elementary problem of calculus of variations. This allows us to establish sufficient optimality conditions for our problems. We establish in the last part of this paper some comparison results for eigenvalues via symmetrization.