Modified Dini functions: monotonicity patterns and functional inequalities

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.