Inequalities for the generalized Marcum Q-function

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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
Issue number1
Publication statusPublished - Sep 1 2008



  • 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

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