Inequalities for the generalized Marcum Q-function

Keywords: Generalized Marcum Q-function Non-central chi and chi-squared distribution Modified Bessel functions Log-concavity NBU property a b s t r a c t In this paper, we consider the generalized Marcum Q-function of order m > 0 real, defined by Q m ða; bÞ ¼ 1 a mÀ1 Z 1 b t m e À t 2 þa 2 2 I mÀ1 ðatÞdt; where a; b P 0, I m 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 ¼ 0. Our aim is to prove that the function m 7 ! Q m ða; bÞ is strictly increasing on ð0; 1Þ for each a P 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.