Inequalities for the generalized Marcum Q-function

Abstract

In this paper, we consider the generalized Marcum Q-function of order ν > 0 real, defined by Q ν ( a , b ) = 1 a ν - 1 ∫ b ∞ t ν e - t 2 + a 2 2 I ν - 1 ( at ) d t , where a , b ⩾ 0 , I ν 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 ν ↦ Q ν ( a , b ) is strictly increasing on ( 0 , ∞ ) for each a ⩾ 0 , b > 0 , and to deduce some interesting inequalities for the function Q ν . 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.