On Turán Type Inequalities for Modified Bessel Functions

In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Turán type inequalities for these functions. Moreover, we present some new Turán type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has an application in the study of stability of radially symmetric solutions in a generalized FitzHugh-Nagumo equation in two spatial dimensions. At the end of this note an open problem is posed, which may be of interest for further research. 1. Some inequalities for modified Bessel functions Let us denote with Iν and Kν the modified Bessel functions of the first and second kind, respectively. For definitions, recurrence formulas and other properties of modified Bessel functions of the first and second kind we refer to the classical book of Watson [35]. In 2007, motivated by a problem which arises in biophysics, Penfold et al. [31] proved that the product of the modified Bessel functions of the first and second kind, 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 [32]. In order to shorten the proof due to Penfold et al. [31], recently the first author [5] pointed out that the Turán type inequalities for modified Bessel functions of the first and second kinds are in fact equivalent to some known inequalities for the logarithmic derivatives of the functions in question. For the reader’s convenience we recall here the historical facts for these Turán type inequalities (see [5, 8] for more details). More precisely, in view of the recurrence relations (1) Iν−1(u) = (ν/u)Iν(u) + I ′ ν(u) and Iν+1(u) = I ′ ν(u)− (ν/u)Iν(u), the Turán type inequality (2) Iν−1(u)Iν+1(u)− [Iν(u)] < 0 Received by the editors April 23, 2010 and, in revised form, June 28, 2011. 2010 Mathematics Subject Classification. Primary 33C10, 39B62.