### 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 |
---|---|

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

- Applied Mathematics
- Computational Mathematics
- Numerical Analysis

### Cite this

*Applied Mathematics and Computation*,

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

**Inequalities for the generalized Marcum Q-function.** / Sun, Yin; Baricz, A.

Research output: Contribution to journal › Article

*Applied Mathematics and Computation*, vol. 203, no. 1, pp. 134-141. https://doi.org/10.1016/j.amc.2008.04.009

}

TY - JOUR

T1 - Inequalities for the generalized Marcum Q-function

AU - Sun, Yin

AU - Baricz, A.

PY - 2008/9/1

Y1 - 2008/9/1

N2 - In this paper, we consider the generalized Marcum Q-function of order ν > 0 real, defined byQν (a, b) = frac(1, aν - 1) ∫b∞ tν e- frac(t2 + a2, 2) Iν - 1 (at) d t,where a, b ≥ 0, Iν 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} Qν (a, b) is strictly increasing on (0, ∞) for each a ≥ 0, b > 0, and to deduce some interesting inequalities for the function Qν. 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.

AB - In this paper, we consider the generalized Marcum Q-function of order ν > 0 real, defined byQν (a, b) = frac(1, aν - 1) ∫b∞ tν e- frac(t2 + a2, 2) Iν - 1 (at) d t,where a, b ≥ 0, Iν 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} Qν (a, b) is strictly increasing on (0, ∞) for each a ≥ 0, b > 0, and to deduce some interesting inequalities for the function Qν. 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.

KW - Generalized Marcum Q-function

KW - Log-concavity

KW - Modified Bessel functions

KW - NBU property

KW - Non-central chi and chi-squared distribution

UR - http://www.scopus.com/inward/record.url?scp=50249098067&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=50249098067&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2008.04.009

DO - 10.1016/j.amc.2008.04.009

M3 - Article

VL - 203

SP - 134

EP - 141

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

IS - 1

ER -