  The generalized Marcum functions $Q_{\mu}(x,y)$ and $P_{\mu}(x,y)$ have as
particular cases the non-central $\chi^2$ and gamma cumulative distributions,
which become central distributions (incomplete gamma function ratios) when the
non-centrality parameter $x$ is set to zero. We analyze monotonicity and
convexity properties for the generalized Marcum functions and for ratios of
Marcum functions of consecutive parameters (differing in one unity) and we
obtain upper and lower bounds for the Marcum functions. These bounds are proven
to be sharper than previous estimations for a wide range of the parameters.
Additionally we show how to build convergent sequences of upper and lower
bounds. The particularization to incomplete gamma functions, together with some
additional bounds obtained for this particular case, lead to combined bounds
which improve previously exiting inequalities.
