### Abstract

In this paper, we consider the generalized Marcum Q-function of order ν > 0 real, defined byQ_{ν} (a, b) = frac(1, a^{ν - 1}) ∫_{b}^{∞} t^{ν} e^{- frac(t2 + a2, 2)} I_{ν - 1} (at) d t,where a, b ≥ 0, I_{ν} stands for the modified Bessel function of the first kind and the right hand side of the above equation is replaced by its limiting value when a = 0. Our aim is to prove that the function ν {mapping} Q_{ν} (a, b) is strictly increasing on (0, ∞) for each a ≥ 0, b > 0, and to deduce some interesting inequalities for the function Q_{ν}. Moreover, we present a somewhat new viewpoint of the generalized Marcum Q-function, by showing that satisfies the new-is-better-than-used (nbu) property, which arises in economic theory.

Original language | English |
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Pages (from-to) | 134-141 |

Number of pages | 8 |

Journal | Applied Mathematics and Computation |

Volume | 203 |

Issue number | 1 |

DOIs | |

Publication status | Published - Sep 1 2008 |

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### Keywords

- Generalized Marcum Q-function
- Log-concavity
- Modified Bessel functions
- NBU property
- Non-central chi and chi-squared distribution

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*Applied Mathematics and Computation*,

*203*(1), 134-141. https://doi.org/10.1016/j.amc.2008.04.009