Bounds for the product of modified Bessel functions

Baricz, Árpád; Maširević, Dragana Jankov; Ponnusamy, Saminathan; Singh, Sanjeev

doi:10.1007/s00010-016-0414-2

Cited by 9 publications

(5 citation statements)

References 11 publications

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“…The inequalities for K ν (x) can be found in [22], whilst the inequality for I ν (x) can be found in [23] and [27], which extends a result of [34]. A survey of related inequalities for modified Bessel functions is given by [7], and lower and upper bounds for the ratios Iν (x) I ν−1 (x) and Kν (x) K ν−1 (x) , which improve on inequalities (A.54) -(A.56), are also given in [22] and [33].…”

Section: A Elementary Properties Of Modified Bessel Functionsmentioning

confidence: 63%

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Inequalities for integrals of modified Bessel functions and expressions involving them

Gaunt

2018

Journal of Mathematical Analysis and Applications

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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, we show that we obtain the best possible constant or that our bounds are tight in certain limits. We apply these inequalities to obtain uniform bounds for several expressions involving integrals of modified Bessel functions. Such expressions occur in Stein's method for variance-gamma approximation, and the results obtained in this paper allow for technical advances in the method. We also present some open problems that arise from this research.

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Section: A Elementary Properties Of Modified Bessel Functionsmentioning

confidence: 63%

“…Part (ii) is proved in Lemma 3 of [16]. See also [5] and [7] for a number of results and upper bounds for the product I ν (x)K ν (x). x ν…”

Section: Uniform Bounds For Expressions Involving Integrals Of Modifimentioning

confidence: 94%

Inequalities for integrals of modified Bessel functions and expressions involving them

Gaunt

2018

Journal of Mathematical Analysis and Applications

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“…ν (x) and positive for the other two roots. Now writing −λ ν (x) 4 = −λ ν (x)λ ν (x) 3 and using (20) to eliminate λ ν (x) 3 we arrive at…”

Section: Resultsmentioning

confidence: 99%

Monotonicity Properties for Ratios and Products of Modified Bessel Functions and Sharp Trigonometric Bounds

Segura

2021

Results Math

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Let $$I_{\nu }(x)$$ I ν ( x ) and $$K_{\nu }(x)$$ K ν ( x ) be the first and second kind modified Bessel functions. It is shown that the nullclines of the Riccati equation satisfied by $$x^{\alpha } \Phi _{i,\nu }(x)$$ x α Φ i , ν ( x ) , $$i=1,2$$ i = 1 , 2 , with $$\Phi _{1,\nu }=I_{\nu -1}(x)/I_{\nu }(x)$$ Φ 1 , ν = I ν - 1 ( x ) / I ν ( x ) and $$\Phi _{2,\nu }(x)=-K_{\nu -1}(x)/K_{\nu }(x)$$ Φ 2 , ν ( x ) = - K ν - 1 ( x ) / K ν ( x ) , are bounds for $$x^{\alpha } \Phi _{i,\nu }(x)$$ x α Φ i , ν ( x ) , which are solutions with unique monotonicity properties; these bounds hold at least for $$\pm \alpha \notin (0,1)$$ ± α ∉ ( 0 , 1 ) and $$\nu \ge 1/2$$ ν ≥ 1 / 2 . Properties for the product $$P_{\nu }(x)=I_{\nu }(x)K_{\nu }(x)$$ P ν ( x ) = I ν ( x ) K ν ( x ) can be obtained as a consequence; for instance, it is shown that $$P_{\nu }(x)$$ P ν ( x ) is decreasing if $$\nu \ge -1$$ ν ≥ - 1 (extending the known range of this result) and that $$xP_{\nu }(x)$$ x P ν ( x ) is increasing for $$\nu \ge 1/2$$ ν ≥ 1 / 2 . We also show that the double ratios $$W_{i,\nu }(x)=\Phi _{i,\nu +1}(x)/\Phi _{i,\nu }(x)$$ W i , ν ( x ) = Φ i , ν + 1 ( x ) / Φ i , ν ( x ) are monotonic and that these monotonicity properties are exclusive of the first and second kind modified Bessel functions. Sharp trigonometric bounds can be extracted from the monotonicity of the double ratios. The trigonometric bounds for the ratios and the product are very accurate as $$x\rightarrow 0^+$$ x → 0 + , $$x\rightarrow +\infty $$ x → + ∞ and $$\nu \rightarrow +\infty $$ ν → + ∞ in the sense that the first two terms in the power series expansions in these limits are exact.

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“…The monotonicity results of Lemma 3.2 for the products K ν+1 (x)L ν (x) and xK ν+2 (x)L ν (x) complement monotonicity results that have been established for the products K ν (x)I ν (x) (see [1,2,29,30]), xK ν (x)I ν (x) (see [21]) and xK ν+1 (x)I ν (x) (see [13]). We also note that a number of bounds for the product K ν (x)I ν (x) have been obtained by [4]. In light of these results, it is natural to ask whether a monotonicity result is available for the product xK ν+1 (x)L ν (x), which is also present in Lemma 3.2.…”

Section: Lemmasmentioning

confidence: 85%

Bounds for an integral involving the modified Struve function of the first kind

Gaunt¹

2021

Preprint

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Simple upper and lower bounds are established for the integralx 0 e −βt t ν Lν (t) dt, where x > 0, ν > −1, 0 < β < 1 and Lν (x) is the modified Struve function of the first kind. These bounds complement and improve on existing results, through either sharper bounds or increased ranges of validity. In deriving our bounds, we obtain some monotonicity results and inequalities for products of the modified Struve function of the first kind and the modified Bessel function of the second kind Kν (x), as well as a new bound for the ratio Lν (x)/L ν−1 (x).

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Bounds for the product of modified Bessel functions

Cited by 9 publications

References 11 publications

Inequalities for integrals of modified Bessel functions and expressions involving them

Inequalities for integrals of modified Bessel functions and expressions involving them

Monotonicity Properties for Ratios and Products of Modified Bessel Functions and Sharp Trigonometric Bounds

Bounds for an integral involving the modified Struve function of the first kind

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