Tight Bounds of the Generalized Marcum Q-Function Based on Log-Concavity

In this paper, we manage to prove the log-concavity of the generalized Marcum Q-function Q_nu(a, b) with respect to its order nu on (1, infin). The proof relies on a powerful mathematical concept named total positivity. Based on the recursion relation of the generalized Marcum Q-function, a new intuitive formula for Q_nu(a,b) is proposed, where nu is an odd multiple of 0.5. After these results, we derive upper and lower bounds for the generalized Marcum Q-function of positive integer order m. Numerical results show that in most of the cases our proposed bounds are much tighter than the existing bounds in the literature. It is surprising to see that the relative errors of the proposed bounds converge to 0 when b approaches infinite.