Monotonicity properties and bounds for the chi-square and gamma distributions

•A sharp analysis of the convexity properties of Marcum functions is performed.•Monotonicity properties for ratios of consecutive Marcum functions are obtained.•The new bounds for Marcum functions improve previous bounds in large regions.•New and improved monotonicity properties are obtained in the central case.•The combined bounds obtained for the central case are superior to previous bounds.

The generalized Marcum functions Qμ(x,y)Qμ(x,y) and Pμ(x,y)Pμ(x,y) have as particular cases the non-central χ2χ2 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 existing inequalities.