On a product of modified Bessel functions

Let Iν and Kν denote the modified Bessel functions of the first and second kinds of order ν . In this note we prove that the monotonicity of u↦ Iν(u)Kν(u) on (0,∞ ) for all ν ≥ -1/2 is an almost immediate consequence of the corresponding Turán type inequalities for the modified Bessel functions of the first and second kinds of order ν . Moreover, we show that the function u↦ Iν(u)Kν(u) is strictly completely monotonic on (0,∞ ) for all ν ∈ [-1/2,1/2]. At the end of this note, a conjecture is stated.