On a product of modified Bessel functions
Author:
Árpád Baricz
Journal:
Proc. Amer. Math. Soc. 137 (2009), 189193
MSC (2000):
Primary 33C10, 33C15
Published electronically:
August 1, 2008
MathSciNet review:
2439440
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Abstract: Let and denote the modified Bessel functions of the first and second kinds of order In this note we prove that the monotonicity of on for all is an almost immediate consequence of the corresponding Turán type inequalities for the modified Bessel functions of the first and second kinds of order Moreover, we show that the function is strictly completely monotonic on for all At the end of this note, a conjecture is stated.
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 Amos, D.E., 1974, Computation of modified Bessel functions and their ratios. Mathematics of Computation, 28, 239251. MR 0333287 (48:11612)
 2.
 Baricz, Á., 2007, Turán type inequalities for generalized complete elliptic integrals. Mathematische Zeithschrift, 256(4), 895911. MR 2308896
 3.
 Baricz, Á., 2008, Turán type inequalities for hypergeometric functions. Proceedings of the American Mathematical Society, 136(9), 32233229.
 4.
 Baricz, Á., 2008, Functional inequalities involving Bessel and modified Bessel functions of the first kind. Expositiones Mathematicae, 26(3), 279293.
 5.
 Baricz, Á., Turán type inequalities for some probability density functions. Studia Scientiarium Mathematicarum Hungarica (submitted).
 6.
 Gronwall, T.H., 1932, An inequality for the Bessel functions of the first kind with imaginary argument. Annals of Mathematics, 33(2), 275278. MR 1503051
 7.
 Ismail, M.E.H. and Muldoon, M.E., 1978, Monotonicity of the zeros of a crossproduct of Bessel functions. SIAM Journal on Mathematical Analysis, 9(4), 759767. MR 0486686 (58:6388)
 8.
 Joshi, C.M. and Bissu, S.K., 1991, Some inequalities of Bessel and modified Bessel functions. Journal of the Australian Mathematical Society (Series A), 50, 333342. MR 1094928 (92d:33007)
 9.
 Laforgia, A. and Natalini, P., 2006, On some Turántype inequalities. Journal of Inequalities and Applications, 2006, Article 29828, 6 pp. MR 2221215 (2007c:26028)
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 Lorch, L., 1994, Monotonicity of the zeros of a cross product of Bessel functions. Methods and Applications of Analysis, 1(1), 7580. MR 1260384 (95b:33008)
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 Miller, K.S. and Samko, G., 2001, Completely monotonic functions. Integral Transforms and Special Functions, 12, 389402. MR 1872377 (2002j:26006)
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 Nåsell, I., 1978, Rational bounds for ratios of modified Bessel functions. SIAM Journal on Mathematical Analysis, 9, 111. MR 0466662 (57:6539)
 13.
 Penfold, R., VandenBroeck, J.M. and Grandison, S., 2007, Monotonicity of some modified Bessel function products. Integral Transforms and Special Functions, 18(2), 139144. MR 2290352 (2007m:33015)
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 Phillips, R.S. and Malin, H., 1950, Bessel function approximations. American Journal of Mathematics, 72, 407418. MR 0035346 (11:720d)
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 Thiruvenkatachar, V.R. and Nanjundiah, T.S., 1951, Inequalities concerning Bessel functions and orthogonal polynomials. Proceedings of the Indian Academy of Sciences. Section A, 33, 373384. MR 0048635 (14:44b)
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Additional Information
Árpád Baricz
Affiliation:
Faculty of Economics, BabeşBolyai University, RO400591 ClujNapoca, Romania
Email:
bariczocsi@yahoo.com
DOI:
http://dx.doi.org/10.1090/S0002993908095713
PII:
S 00029939(08)095713
Keywords:
Modified Bessel functions,
complete monotonicity,
Tur\'an type inequalities.
Received by editor(s):
December 13, 2007
Published electronically:
August 1, 2008
Additional Notes:
This research was partially supported by the Institute of Mathematics, University of Debrecen, Hungary
Dedicated:
Dedicated to my son Koppány
Communicated by:
Carmen C. Chicone
Article copyright:
© Copyright 2008
American Mathematical Society
