On Turán type inequalities for modified Bessel functions

Baricz, Árpád; Ponnusamy, Saminathan

doi:10.1090/S0002-9939-2012-11325-5

On Turán type inequalities for modified Bessel functions
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by Árpád Baricz and Saminathan Ponnusamy PDF

Proc. Amer. Math. Soc. 141 (2013), 523-532 Request permission

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.

References

D. E. Amos, Computation of modified Bessel functions and their ratios, Math. Comp. 28 (1974), 239–251. MR 333287, DOI 10.1090/S0025-5718-1974-0333287-7
Árpád Baricz, Turán type inequalities for generalized complete elliptic integrals, Math. Z. 256 (2007), no. 4, 895–911. MR 2308896, DOI 10.1007/s00209-007-0111-x
Árpád Baricz, Turán type inequalities for hypergeometric functions, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3223–3229. MR 2407087, DOI 10.1090/S0002-9939-08-09353-2
Árpád Baricz, Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math. 26 (2008), no. 3, 279–293. MR 2437098, DOI 10.1016/j.exmath.2008.01.001
Árpád Baricz, On a product of modified Bessel functions, Proc. Amer. Math. Soc. 137 (2009), no. 1, 189–193. MR 2439440, DOI 10.1090/S0002-9939-08-09571-3
Árpád Baricz, Tight bounds for the generalized Marcum $Q$-function, J. Math. Anal. Appl. 360 (2009), no. 1, 265–277. MR 2548382, DOI 10.1016/j.jmaa.2009.06.055
Á. Baricz, Turán type inequalities for some probability density functions, Studia Scientiarium Mathematicarum Hungarica 47 (2010) 175–189.
Árpád Baricz, Turán type inequalities for modified Bessel functions, Bull. Aust. Math. Soc. 82 (2010), no. 2, 254–264. MR 2685149, DOI 10.1017/S000497271000002X
Salomon Bochner, Harmonic analysis and the theory of probability, University of California Press, Berkeley-Los Angeles, Calif., 1955. MR 0072370, DOI 10.1525/9780520345294
P.E. Cantrell, On the calculation of the generalized $Q-$function via Parl’s method, IEEE Transactions on Information Theory 32(6) (1986) 817–824.
James Alan Cochran, The monotonicity of modified Bessel functions with respect to their order, J. Math. and Phys. 46 (1967), 220–222. MR 213624, DOI 10.1002/sapm1967461220
Arjen Doelman, Peter van Heijster, and Tasso J. Kaper, Pulse dynamics in a three-component system: existence analysis, J. Dynam. Differential Equations 21 (2009), no. 1, 73–115. MR 2482009, DOI 10.1007/s10884-008-9125-2
T. H. Gronwall, An inequality for the Bessel functions of the first kind with imaginary argument, Ann. of Math. (2) 33 (1932), no. 2, 275–278. MR 1503051, DOI 10.2307/1968329
Philip Hartman, Completely monotone families of solutions of $n$-th order linear differential equations and infinitely divisible distributions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 2, 267–287. MR 404760
Philip Hartman, On the products of solutions of second order disconjugate differentialequations and the Whittaker differential equation, SIAM J. Math. Anal. 8 (1977), no. 3, 558–571. MR 435510, DOI 10.1137/0508044
Philip Hartman and Geoffrey S. Watson, “Normal” distribution functions on spheres and the modified Bessel functions, Ann. Probability 2 (1974), 593–607. MR 370687, DOI 10.1214/aop/1176996606
P. van Heijster, Front Interactions in a Three-Component System, Ph.D. Thesis, Universiteit van Amsterdam, Amsterdam, 2009.
Peter van Heijster, Arjen Doelman, and Tasso J. Kaper, Pulse dynamics in a three-component system: stability and bifurcations, Phys. D 237 (2008), no. 24, 3335–3368. MR 2477026, DOI 10.1016/j.physd.2008.07.014
P. van Heijster, A. Doelman, T. J. Kaper, and K. Promislow, Front interactions in a three-component system, SIAM J. Appl. Dyn. Syst. 9 (2010), no. 2, 292–332. MR 2644922, DOI 10.1137/080744785
Peter van Heijster and Björn Sandstede, Planar radial spots in a three-component FitzHugh-Nagumo system, J. Nonlinear Sci. 21 (2011), no. 5, 705–745. MR 2841985, DOI 10.1007/s00332-011-9098-x
Mourad E. H. Ismail, Integral representations and complete monotonicity of various quotients of Bessel functions, Canadian J. Math. 29 (1977), no. 6, 1198–1207. MR 463527, DOI 10.4153/CJM-1977-119-5
Mourad E. H. Ismail, Bessel functions and the infinite divisibility of the Student $t$-distribution, Ann. Probability 5 (1977), no. 4, 582–585. MR 448480, DOI 10.1214/aop/1176995766
Mourad E. H. Ismail and Martin E. Muldoon, Monotonicity of the zeros of a cross-product of Bessel functions, SIAM J. Math. Anal. 9 (1978), no. 4, 759–767. MR 486686, DOI 10.1137/0509055
C. M. Joshi and S. K. Bissu, Some inequalities of Bessel and modified Bessel functions, J. Austral. Math. Soc. Ser. A 50 (1991), no. 2, 333–342. MR 1094928, DOI 10.1017/S1446788700032791
A. Laforgia and P. Natalini, On some Turán-type inequalities, J. Inequal. Appl. , posted on (2006), Art. ID 29828, 6. MR 2221215, DOI 10.1155/JIA/2006/29828
Andrea Laforgia and Pierpaolo Natalini, Some inequalities for modified Bessel functions, J. Inequal. Appl. , posted on (2010), Art. ID 253035, 10. MR 2592860, DOI 10.1155/2010/253035
Lee Lorch, Monotonicity of the zeros of a cross product of Bessel functions, Methods Appl. Anal. 1 (1994), no. 1, 75–80. MR 1260384, DOI 10.4310/MAA.1994.v1.n1.a6
Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968, DOI 10.1007/978-3-662-11761-3
K. S. Miller and S. G. Samko, Completely monotonic functions, Integral Transform. Spec. Funct. 12 (2001), no. 4, 389–402. MR 1872377, DOI 10.1080/10652460108819360
Martin E. Muldoon, Convexity properties of special functions and their zeros, Recent progress in inequalities (Niš, 1996) Math. Appl., vol. 430, Kluwer Acad. Publ., Dordrecht, 1998, pp. 309–323. MR 1609967
Robert Penfold, Jean-Marc Vanden-Broeck, and Scott Grandison, Monotonicity of some modified Bessel function products, Integral Transforms Spec. Funct. 18 (2007), no. 1-2, 139–144. MR 2290352, DOI 10.1080/10652460601041219
R. S. Phillips and Henry Malin, Bessel function approximations, Amer. J. Math. 72 (1950), 407–418. MR 35346, DOI 10.2307/2372042
Iosif Pinelis, L’Hospital type results for monotonicity, with applications, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 1, Article 5, 5. MR 1888920
V. R. Thiruvenkatachar and T. S. Nanjundiah, Inequalities concerning Bessel functions and orthogonal polynomials, Proc. Indian Acad. Sci., Sect. A. 33 (1951), 373–384. MR 0048635, DOI 10.1007/BF03178130
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923