A note on Lyapunov-type inequalities for fractional boundary value problems with Sturm-Liouville boundary conditions

In this note, we consider fractional Strum-Liouville boundary value problem containing Caputo derivative of order $\alpha$, $ 1<\alpha\leq 2$ with mixed boundary conditions. We establish Cauchy-Schwarz-type inequality to determine a lower bound for the smallest eigenvalues. We give comparison between the smallest eigenvalues and its lower bounds obtained from the Lyapunov-type and Cauchy-Schwarz-type inequalities. Result shows that the Lyapunov-type inequality gives the worse and Cauchy-Schwarz-type inequality gives better lower bound estimates for the smallest eigenvalues. We then use these inequalities to obtain an interval where a linear combination of certain Mittag-Leffler functions have no real zeros.