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. 1. Preliminaries and main results Let Iν and Kν 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 → Pν(u) = Iν(u)Kν(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 → Pν(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 → Iν(u)Kν(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 have (−1) [Iν(u)Kν(u)] > 0. In order to achieve our goal we improve some of the results of Phillips and Malin [14, Eq. 1] concerning bounds for the logarithmic derivatives of the modified Bessel and Hankel functions. Our main result reads as follows: Theorem 1. The following assertions are true: a. the function u → Pν(u) is strictly decreasing on (0,∞) for all ν ≥ −1/2; Received by the editors December 13, 2007. 2000 Mathematics Subject Classification. Primary 33C10, 33C15.