Lyapunov-type inequality for fractional boundary value problems with Hilfer derivative

In this work, we establish Lyapunov-type inequalities for fractional boundary value problems containing Hilfer derivative of order α , 1 < α 2 and type 0 β 1 .We consider the boundary value problems with the Dirichlet, and a mixed set of Dirichlet and Neumann boundary conditions.We consider both integer and fractional order eigenvalue problems, determine a lower bound for the smallest eigenvalue using Lyapunov-type inequalities, and improve these bounds using semi maximum norm and Cauchy-Schwarz inequalities.We use the improved lower bounds to obtain intervals where a certain Mittag-Leffler functions have no real zeros.Further, we discuss the particular cases for the type β = 0 and β = 1 , which give the results respectively for Riemann-Liouville and Caputo fractional boundary value as well as eigenvalue problems.For both the fractional and the integer order eigenvalue problems, we give a comparison between the smallest eigenvalue and its lower bounds obtained from the Lyapunov-type, semi maximum norm and Cauchy-Schwarz inequalities.Results show that the Lyapunov-type inequality gives the worse and semi maximum norm and Cauchy-Schwarz inequalities give better lower bound estimates for the smallest eigenvalues.