Inequalities for the generalized Marcum Q-function

In this paper, we consider the generalized Marcum Q-function of order v > 0 real, defined by Q(v)(a,b) =1/a(v-1)integral(infinity)(b) t(v)e - t(2)+a(2)/2 I(v-1) (at)dt, where a,b >= 0, I(v) stands for the modified Bessel function of the first kind and the right hand side of the above equation is replaced by its limiting value when a = 0. Our aim is to prove that the function v -> Q(v)(a,b) is strictly increasing on (0, infinity) for each a >= 0, b > 0, and to deduce some interesting inequalities for the function Q m. Moreover, we present a somewhat new viewpoint of the generalized Marcum Q-function, by showing that satisfies the new-is-better-than-used (nbu) property, which arises in economic theory. (c) 2008 Elsevier Inc. All rights reserved.