Inequalities for the generalized Marcum Q-function

In this paper, we consider the generalized Marcum Q  -function of order ν>0ν>0 real, defined byQν(a,b)=1aν-1∫b∞tνe-t2+a22Iν-1(at)dt,where a,b⩾0a,b⩾0, IνIν 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=0a=0. Our aim is to prove that the function ν↦Qν(a,b) is strictly increasing on (0,∞) for each a⩾0a⩾0, b>0b>0, and to deduce some interesting inequalities for the function QνQν. 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.