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

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