Bounds 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 ) dt , where a > 0, b ⩾ 0 and I ν stands for the modified Bessel function of the first kind. Our aim is to extend some results on the (first order) Marcum Q-function to the generalized Marcum Q-function in order to deduce some new lower and upper bounds. Moreover, we show that the proposed bounds are very tight for the generalized Marcum Q-function of integer order, and we deduce some new inequalities for the more general case of real order. The chief tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind, which are based on a criterion for the monotonicity of the quotient of two Maclaurin series.