  We find upper and lower bounds for the first eigenvalue and the volume
entropy of a noncompact real analytic K\"ahler manifold, in terms of Calabi's
diastasis function and diastatic entropy, which are sharp in the case of the
complex hyperbolic space. As a corollary we obtain explicit lower bounds for
the first eigenvalue of the geodesic balls of an Hermitian symmetric space of
noncompact type.
