  In this paper, we present a comprehensive study of the monotonicity and
log-concavity of the generalized Marcum and Nuttall Q-functions. More
precisely, a simple probabilistic method is firstly given to prove the
monotonicity of these two functions. Then, the log-concavity of the generalized
Marcum Q-function and its deformations is established with respect to each of
the three parameters. Since the Nuttall Q-function has similar probabilistic
interpretations as the generalized Marcum Q-function, we deduce the
log-concavity of the Nuttall Q-function. By exploiting the log-concavity of
these two functions, we propose new tight lower and upper bounds for the
generalized Marcum and Nuttall Q-functions. Our proposed bounds are much
tighter than the existing bounds in the literature in most of the cases. The
relative errors of our proposed bounds converge to 0 as b tends to infinity.
The numerical results show that the absolute relative errors of the proposed
bounds are less than 5% in most of the cases. The proposed bounds can be
effectively applied to the outage probability analysis of interference-limited
systems such as cognitive radio and wireless sensor network, in the study of
error performance of various wireless communication systems operating over
fading channels and extracting the log-likelihood ratio for differential
phase-shift keying (DPSK) signals.
