  We deduce some new functional inequalities, like Tur\'an type inequalities,
Redheffer type inequalities, and a Mittag-Leffler expansion for a special
combination of modified Bessel functions of the first kind, called modified
Dini functions. Moreover, we show the complete monotonicity of a quotient of
modified Dini functions by introducing a new continuous infinitely divisible
probability distribution. The key tool in our proofs is a recently developed
infinite product representation for a special combination of Bessel functions
of the first, which was very useful in determining the radius of convexity of
some normalized Bessel functions of the first kind.
