Tight bounds for the generalized Marcum Q-function

Abstract

In this paper we study the generalized Marcum Q-function of order $\nu >0$ real, defined by

$$Q_{\nu }(a,b)=\frac{1}{a^{\nu -1}}\underset{b}{\overset{\infty }{\int }}{t}^{\nu }{e}^{-\frac{t^2+a^2}{2}}{I}_{\nu -1}(at)\mathrm{dt},$$

where $a>0$, $b⩾0$ and $I_{\nu }$ stands for the modified Bessel function of the first kind. Our aim is to improve and extend some recent results of Wang to the generalized Marcum Q-function in order to deduce some sharp lower and upper bounds. In both cases $b⩾a$ and $b<a$ we give the best possible upper bound for $Q_{\nu }(a,b)$. The key tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind. These monotonicity properties are deduced from some results on modified Bessel functions, which have been used in wave mechanics and finite elasticity.