  In this note our aim is to present some monotonicity properties of the
product of modified Bessel functions of first and second kind. Certain bounds
for the product of modified Bessel functions of first and second kind are also
obtained. These bounds improve and extend known bounds for the product of
modified Bessel functions of first and second kind of order zero. A new Tur\'an
type inequality is also given for the product of modified Bessel functions, and
some open problems are stated, which may be of interest for further research.
