Inequalities for the generalized Marcum Q-function

Yin Sun, A. Baricz

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)134-141
Number of pages8
JournalApplied Mathematics and Computation
Volume203
Issue number1
DOIs
Publication statusPublished - Sep 1 2008

Fingerprint

Q-function
Bessel function of the first kind
Modified Bessel Functions
Bessel functions
Deduce
Strictly
Limiting
Economics

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

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

In: Applied Mathematics and Computation, Vol. 203, No. 1, 01.09.2008, p. 134-141.

Research output: Contribution to journalArticle

@article{648b1384cc154688b03fd1781572b2db,
title = "Inequalities for the generalized Marcum Q-function",
abstract = "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.",
keywords = "Generalized Marcum Q-function, Log-concavity, Modified Bessel functions, NBU property, Non-central chi and chi-squared distribution",
author = "Yin Sun and A. Baricz",
year = "2008",
month = "9",
day = "1",
doi = "10.1016/j.amc.2008.04.009",
language = "English",
volume = "203",
pages = "134--141",
journal = "Applied Mathematics and Computation",
issn = "0096-3003",
publisher = "Elsevier Inc.",
number = "1",

}

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 -