### Abstract

In this paper we study the generalized Marcum Q-function of order ν > 0 real, defined byQ_{ν} (a, b) = frac(1, a^{ν - 1}) underover(∫, b, ∞) t^{ν} e^{- frac(t2 + a2, 2)} I_{ν - 1} (a t) dt, where a > 0, b ≥ 0 and I_{ν} stands for the modified Bessel function of the first kind. Our aim is to improve and extend some recent results of Wang to the generalized Marcum Q-function in order to deduce some sharp lower and upper bounds. In both cases b ≥ a and b <a we give the best possible upper bound for Q_{ν} (a, b). The key tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind. These monotonicity properties are deduced from some results on modified Bessel functions, which have been used in wave mechanics and finite elasticity.

Original language | English |
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Pages (from-to) | 265-277 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 360 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1 2009 |

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### Keywords

- Complementary error function
- Generalized Marcum Q-function
- Lower and upper bounds
- Marcum Q-function
- Modified Bessel functions
- Sharp bounds

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Tight bounds for the generalized Marcum Q-function.** / Baricz, A.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 360, no. 1, pp. 265-277. https://doi.org/10.1016/j.jmaa.2009.06.055

}

TY - JOUR

T1 - Tight bounds for the generalized Marcum Q-function

AU - Baricz, A.

PY - 2009/12/1

Y1 - 2009/12/1

N2 - In this paper we study the generalized Marcum Q-function of order ν > 0 real, defined byQν (a, b) = frac(1, aν - 1) underover(∫, b, ∞) tν e- frac(t2 + a2, 2) Iν - 1 (a t) dt, where a > 0, b ≥ 0 and Iν stands for the modified Bessel function of the first kind. Our aim is to improve and extend some recent results of Wang to the generalized Marcum Q-function in order to deduce some sharp lower and upper bounds. In both cases b ≥ a and b ν (a, b). The key tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind. These monotonicity properties are deduced from some results on modified Bessel functions, which have been used in wave mechanics and finite elasticity.

AB - In this paper we study the generalized Marcum Q-function of order ν > 0 real, defined byQν (a, b) = frac(1, aν - 1) underover(∫, b, ∞) tν e- frac(t2 + a2, 2) Iν - 1 (a t) dt, where a > 0, b ≥ 0 and Iν stands for the modified Bessel function of the first kind. Our aim is to improve and extend some recent results of Wang to the generalized Marcum Q-function in order to deduce some sharp lower and upper bounds. In both cases b ≥ a and b ν (a, b). The key tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind. These monotonicity properties are deduced from some results on modified Bessel functions, which have been used in wave mechanics and finite elasticity.

KW - Complementary error function

KW - Generalized Marcum Q-function

KW - Lower and upper bounds

KW - Marcum Q-function

KW - Modified Bessel functions

KW - Sharp bounds

UR - http://www.scopus.com/inward/record.url?scp=68049137315&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=68049137315&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2009.06.055

DO - 10.1016/j.jmaa.2009.06.055

M3 - Article

AN - SCOPUS:68049137315

VL - 360

SP - 265

EP - 277

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -