Turán type inequalities for hypergeometric functions

Baricz, Árpád

doi:10.1090/s0002-9939-08-09353-2

Cited by 53 publications

(55 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For more details on Turán type inequalities the interested reader is referred to the papers [2], [3], [4], [5], [8], [9], [10], [15] and to the references therein. Finally, let us note that in 1994 Lorch [10, p. 79] proved that (2.3) in fact holds for all ν ≥ −1/2 and u > 0.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

On a product of modified Bessel functions

Baricz

2008

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. Let I ν and K ν denote the modified Bessel functions of the first and second kinds of order ν. In this note we prove that the monotonicity of u → I ν (u)K ν (u) on (0, ∞) for all ν ≥ −1/2 is an almost immediate consequence of the corresponding Turán type inequalities for the modified Bessel functions of the first and second kinds of order ν. Moreover, we show that the function u →At the end of this note, a conjecture is stated. Preliminaries and main resultsLet I ν and K ν denote, as usual, the modified Bessel functions of the first and second kinds of order ν. Recently, motivated by a problem which arises in biophysics, Penfold et al. [13, Theorem 3.1] proved, in a complicated way, that the product of the modified Bessel functions of the first and second kinds, i.e. u → P ν (u) = I ν (u)K ν (u), is strictly decreasing on (0, ∞) for all ν ≥ 0. It is worth mentioning that this result for ν = n ≥ 1, a positive integer, was verified in 1950 by Phillips and Malin [14, Corollary 2.2]. In this note our aim is to show that using the idea of Phillips and Malin, the monotonicity of u → P ν (u) for ν ≥ −1/2 can be verified easily by using the corresponding Turán type inequalities for modified Bessel functions. Moreover, we show that the function u → I ν (u)K ν (u) is strictly completely monotonic on (0, ∞) for all ν ∈ [−1/2, 1/2], i.e. for all u > 0, ν ∈ [−1/2, 1/2] and m = 0, 1, 2, . . . , we haveIn order to achieve our goal we improve some of the results of Phillips and Malin

show abstract

Section: Proof Of the Main Resultsmentioning

confidence: 99%

On a product of modified Bessel functions

Baricz

2008

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is also interesting to compare these results with the elegant Theorem 2.3 and open problems in [5] regarding Turán-type and arithmetic mean-geometric mean inequalities involving the Gaussian hypergeometric function 2 F 1 .…”

Section: Corollary 3 Suppose ν ∈ N and A B ν Then For All Nonzero mentioning

confidence: 93%

A note on Turán type and mean inequalities for the Kummer function

Barnard

Gordy

Richards

2009

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

“…Moreover, inequality (8) improves a recently deduced inequality by the first author [6], which is useful in the study of the generalized Marcum Q−function applied frequently in radar signal processing. See [6,26] for more details and also [2,3,4,7,8,24,25,27,34] for more details on Turán type inequalities.…”

Section: Some Inequalities For Modified Bessel Functionsmentioning

confidence: 99%

On Turán type inequalities for modified Bessel functions

Baricz

Ponnusamy

2012

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Turán type inequalities for these functions. Moreover, we present some new Turán type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has an application in the study of stability of radially symmetric solutions in a generalized FitzHugh-Nagumo equation in two spatial dimensions. At the end of this note an open problem is posed, which may be of interest for further research.

show abstract

Turán type inequalities for hypergeometric functions

Abstract: Abstract. In this note our aim is to establish a Turán type inequality for Gaussian hypergeometric functions. This result completes the earlier result that G. Gasper proved for Jacobi polynomials. Moreover, at the end of this note we present some open problems.

Cited by 53 publications

References 18 publications

On a product of modified Bessel functions

On a product of modified Bessel functions

A note on Turán type and mean inequalities for the Kummer function

On Turán type inequalities for modified Bessel functions

Contact Info

Product

Resources

About