Abstract. Let I ν and K ν denote the modified Bessel functions of the first and second kinds of order ν. In this note we prove that the monotonicity of u → I ν (u)K ν (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 →At the end of this note, a conjecture is stated.
Preliminaries and main resultsLet I ν and K ν denote, as usual, the modified Bessel functions of the first and second kinds of order ν. Recently, motivated by a problem which arises in biophysics, Penfold et al. [13, Theorem 3.1] proved, in a complicated way, that the product of the modified Bessel functions of the first and second kinds, i.e. u → P ν (u) = I ν (u)K ν (u), is strictly decreasing on (0, ∞) for all ν ≥ 0. It is worth mentioning that this result for ν = n ≥ 1, a positive integer, was verified in 1950 by Phillips and Malin [14, Corollary 2.2]. In this note our aim is to show that using the idea of Phillips and Malin, the monotonicity of u → P ν (u) for ν ≥ −1/2 can be verified easily by using the corresponding Turán type inequalities for modified Bessel functions. Moreover, we show that the function u → I ν (u)K ν (u) is strictly completely monotonic on (0, ∞) for all ν ∈ [−1/2, 1/2], i.e. for all u > 0, ν ∈ [−1/2, 1/2] and m = 0, 1, 2, . . . , we haveIn order to achieve our goal we improve some of the results of Phillips and Malin