  Motivated by some applications in applied mathematics, biology, chemistry,
physics and engineering sciences, new tight Tur\'an type inequalities for
modified Bessel functions of the first and second kind are deduced. These
inequalities provide sharp lower and upper bounds for the Tur\'anian of
modified Bessel functions of the first and second kind, and in most cases the
relative errors of the bounds tend to zero as the argument tends to infinity.
The chief tools in our proofs are some ideas of Gronwall [19] on ordinary
differential equations, an integral representation of Ismail [28,29] for the
quotient of modified Bessel functions of the second kind and some results of
Hartman and Watson [24,26,59]. As applications of the main results some sharp
Tur\'an type inequalities are presented for the product of modified Bessel
functions of the first and second kind and it is shown that this product is
strictly geometrically concave.

Classical Analysis and ODEs