  In this paper we study the Dini functions and the cross-product of Bessel
functions. Moreover, we are interested on the monotonicity patterns for the
cross-product of Bessel and modified Bessel functions. In addition, we deduce
Redheffer-type inequalities, and the interlacing property of the zeros of Dini
functions and the cross-product of Bessel and modified Bessel functions. Bounds
for logarithmic derivatives of these functions are also derived. The key tools
in our proofs are some recently developed infinite product representations for
Dini functions and cross-product of Bessel functions.
