A new type of sharp bounds for ratios of modified Bessel functions

Ruiz-Antolín, Diego; Segura, Javier

doi:10.1016/j.jmaa.2016.06.011

Cited by 31 publications

(42 citation statements)

References 14 publications

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“…x tanh(x) where both the lower and upper bounds are valid for ν ≥ 1 2 and we have equality if and only if ν = 1 2 , and by [60] (see also [1,39,59]) that, for x > 0, x…”

Section: Boundingmentioning

confidence: 97%

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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“…x tanh(x) where both the lower and upper bounds are valid for ν ≥ 1 2 and we have equality if and only if ν = 1 2 , and by [60] (see also [1,39,59]) that, for x > 0, x…”

Section: Boundingmentioning

confidence: 97%

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

Gaunt

2020

Journal of Mathematical Analysis and Applications

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“…But Theorem 2 of [19] (see also [18] and [13]) states that, for ν > 3 2 − n and x > 0, K ν+n−2 (x) K ν+n−1 (x) > x ν + n − 3/2 + x 2 + (ν + n − 3/2) 2 .…”

Section: A An Inequality Involving the Modified Bessel Function Of Thmentioning

confidence: 99%

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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“…An analogous relation for the ratio I ν (x)/I ν−1 (x) has been used by [1,38,39] to obtain a sequence of iteratively refined upper and lower bounds that converge to the ratio I ν (x)/I ν−1 (x). We do not undertake such an investigation in this paper, and we contend ourselves with the following simple illustration of the approach.…”

Section: )mentioning

confidence: 99%

“…These ratios are also key computational tools in the construction of numerical algorithms for computing modified Bessel functions (see, for example, Algorithms 12.6 and 12.7 of [13]). There is now an extensive literature on lower and upper bounds for these ratios; see [1,5,14,15,17,18,19,26,27,31,32,36,38,39,40,43]. There is also a considerable literature on lower and upper bounds for the ratios I ν (x)/I ν (y) and K ν (x)/K ν (y); see [1,2,4,10,11,17,20,21,24,25,35,37,42], which has been used, for example, to obtain tight bounds for the generalized Marcum Q-function, which arises in radar signal processing [2,10].…”

Section: Introductionmentioning

confidence: 99%