The Weighted Lyapunov-type Inequality And The Extremal Problems Of Eigenvalues

In this paper,we consider the Sturm-Liouville problem with Dirichlet boundary conditions.The main results are as follows:In the second chapter,applying the Green’s funtions and Mercer theorem,we prove a weighted Lyapunov-type inequality,which is a extension of the classical Lyapunov inequality.Secondly,we obtain the optimal constant of the weighted Lyapunov-type inequality and prove that is sharp in the meaning of distribution.In the third chapter,we get the the formula of extremum of the integral of potentials in weighted integral space by using the above extened inequality where one of the eigenvalues is given.Furthermore,we give the extremum of the n-th eigenvalue for the potentials on a closed ball in weighted integrable spaces.