  In this short note, we present the sharp estimate for the existence of a
unique solution for a Hadamard-type fractional differential equations with
two-point boundary value conditions. The method of analysis is obtained by
using the integral of the Green's function and the Banach contraction
principle. Further, we will also obtain a sharper lower bound of the
eigenvalues for an eigenvalue problem. Two examples are presented to clarify
the applicability of the essential results.
