Inequalities for modified Bessel functions and their integrals

Gaunt, Robert E.

doi:10.1016/j.jmaa.2014.05.083

Cited by 72 publications

(97 citation statements)

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“…Similar inequalities for integrals involving the modified Struve function of the first kind L ν (x) have also been established by [9]. The bounds of [7,8] were required in the development of Stein's method [21,4,15] for variance-gamma approximation [5,6,10]. Although, the combination of their simple form and accuracy mean that the inequalities may also prove useful in other problems involving modified Bessel functions; see, for example, [2,3] in which inequalities for the modified Bessel function of the first kind are used to obtain tight bounds for the generalized Marcum Q-function, which frequently arises in radar signal processing.…”

Section: Introductionsupporting

confidence: 59%

Inequalities for some integrals involving modified Bessel functions

Gaunt

2019

Proc. Amer. Math. Soc.

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Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases these inequalities are tight in certain limits. As a consequence, we deduce a tight double inequality, involving the modified Bessel function of the first kind, for a generalized hypergeometric function. We also present some open problems that arise from this research.

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Section: Introductionsupporting

confidence: 59%

Inequalities for some integrals involving modified Bessel functions

Gaunt

2019

Proc. Amer. Math. Soc.

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“…, [19,20,22]. These bounds are given in terms of the modified Bessel function of the first kind and a number of both the lower and upper bounds are tight in the limits x ↓ 0 and x → ∞.…”

Section: Boundingmentioning

confidence: 94%

Bounds for modified Lommel functions of the first kind and their ratios

Gaunt

2020

Journal of Mathematical Analysis and Applications

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The modified Lommel function t µ,ν (x) is an important special function, but to date there has been little progress on the problem of obtaining functional inequalities for t µ,ν (x). In this paper, we advance the literature substantially by obtaining a simple two-sided inequality for the ratio t µ,ν (x)/t µ−1,ν−1 (x) in terms of the ratio I ν (x)/I ν−1 (x) of modified Bessel functions of the first kind, thereby allowing one to exploit the extensive literature on bounds for this ratio. We apply this result to obtain two-sided inequalities for the condition numbers xt ′ µ,ν (x)/t µ,ν (x), the ratio t µ,ν (x)/t µ,ν (y) and the modified Lommel function t µ,ν (x) itself that are given in terms of xI ′ ν (x)/I ν (x), I ν (x)/I ν (y) and I ν (x), respectively, which again allows one to exploit the substantial literature on bounds for these quantities. The bounds obtained in this paper are quite accurate and often tight in certain limits. As an important special case we deduce bounds for modified Struve functions of the first kind and their ratios, some of which are new, whilst others extend the range of validity of some results given in the recent literature.

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“…, is closely related to the modified Bessel function I ν (x), and either shares or has a close analogue to the properties of I ν (x) that were used by [8,10,13] to obtain inequalities for the integrals in (1.1). The function L ν (x) is itself a widely used special function; see a standard reference, such as [17], for its basic properties.…”

Section: Introductionmentioning

confidence: 99%

“…In a series of recent papers [8,10,13], simple lower and upper bounds, involving the modified Bessel function of the first kind I ν (x), were obtained for the integrals x 0 e −γt t ±ν I ν (t) dt, (1.1) where x > 0, 0 ≤ γ < 1 and ν > − 1 2 . For γ = 0 there does not exist simple closed form expressions for these integrals.…”

Section: Introductionmentioning

confidence: 99%