SHARP ESTIMATES FOR THE EIGENVALUES OF SOME DIFFERENTIAL-EQUATIONS

We present optimal upper and lower bounds for the eigenvalues of the d ifferential 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 cond itions when the coefficient functions q(x) and rho(x) are nonnegative and are subjected to some kind of additional constraints. One of the b asic ideas used in our work consists in reducing the problem of maximi zing 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.