On the Lyapunov-type inequalities of a three-point boundary value problem for third order linear differential equations

In this paper, by using Green’s function for second order differential equations with Dirichlet boundary condition, we establish new Lyapunov-type inequalities for third order linear differential equation which improve all existing results in the literature. As an application, we obtain sharp lower bound for the eigenvalues of corresponding equations and for the distance between the end points in the three consecutive zeros of the solution of the equations.