On Tur\'an 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 Tur\'an type inequalities for these functions. Moreover, we present some new Tur\'an type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has 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 a conjecture is posed, which may be of interest for further research.