In this paper, we obtain Hartman-Wintner and Lyapunov-type inequalities for the three-point fractional boundary value problem of the fractional Liouville-Caputo differential equation of order α ∈ (2, 3]. The results presented in this work are sharper than the existing results in the literature. As an application of the results, the fractional Sturm–Liouville eigenvalue problems have also been presented. Moreover, we examine the nonexistence of the nontrivial solution of the fractional boundary value problem.