In this paper, we will introduce some new results about Lyapunov type inequalities by studying the given fractional order Sturm-Liouville problems and fractional order Hamiltonian systems. We state and prove Lyapunov type inequalities for fractional S-L problems equipped by fractional Riemann- Liouville derivatives. In this paper for establishing fractional Hamiltonian systems we will apply recently introduced so called fractional Conformable operators. At the end as applications of these inequalities, some interesting spectral properties for fractional order S-L problems together with stability of fractional Hamiltonian systems will be concluded.