Abstract :
In this paper, we study the generalized Marcum Q-function Q nu(a, b), where a, nu > 0 and b ges 0. Our aim is to extend the results of Corazza and Ferrari (IEEE Trans. Inf. Theory, vol. 48, pp. 3003-3008, 2002) to the generalized Marcum Q-function in order to deduce some new tight lower and upper bounds. The key tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind and some classical inequalities, i.e., the Cauchy-Buniakowski-Schwarz and Chebyshev integral inequalities. These bounds are shown to be very tight for large b, i.e., the relative errors of our bounds converge to zero as b increases. Both theoretical analysis and numerical results are provided to show the tightness of our bounds.
Keywords :
Bessel functions; function evaluation; functional analysis; information theory; integral equations; Bessel function; Cauchy-Buniakowski-Schwarz-Chebyshev integral inequality; generalized Marcum Q-function; monotonicity property; Chebyshev approximation; Chemistry; Digital communication; Elasticity; Integral equations; Laboratories; Mathematics; Physics; Quantum mechanics; Upper bound; Cauchy–Buniakowski–Schwarz and Chebyshev integral inequalities; generalized Marcum $Q$-function; lower and upper bounds; modified Bessel functions of the first kind;