  In this note we offer some log-concavity properties of certain functions
related to Bessel functions of the first kind and modified Bessel functions of
the first and second kind, by solving partially a recent conjecture on the
log-convexity/log-concavity properties for modified Bessel functions of the
first kind and their derivatives. Moreover, we give an application of the
mentioned results by extending two inequalities of van der Corput to Bessel and
modified Bessel functions of the first kind. Similar inequalities are proved
also for modified Bessel functions of the second kind, as well as for
log-concave probability density functions.
