On a product of modified Bessel functions

Author:
Árpád Baricz

Journal:
Proc. Amer. Math. Soc. **137** (2009), 189-193

MSC (2000):
Primary 33C10, 33C15

DOI:
https://doi.org/10.1090/S0002-9939-08-09571-3

Published electronically:
August 1, 2008

MathSciNet review:
2439440

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**Amos, D.E., 1974, Computation of modified Bessel functions and their ratios.*Mathematics of Computation,***28**, 239-251. MR**0333287 (48:11612)****2.**Baricz, Á., 2007, Turán type inequalities for generalized complete elliptic integrals.*Mathematische Zeithschrift,***256**(4), 895-911. MR**2308896****3.**Baricz, Á., 2008, Turán type inequalities for hypergeometric functions.*Proceedings of the American Mathematical Society,***136**(9), 3223-3229.**4.**Baricz, Á., 2008, Functional inequalities involving Bessel and modified Bessel functions of the first kind.*Expositiones Mathematicae*,**26**(3), 279-293.**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), 275-278. MR**1503051****7.**Ismail, M.E.H. and Muldoon, M.E., 1978, Monotonicity of the zeros of a cross-product of Bessel functions.*SIAM Journal on Mathematical Analysis,***9**(4), 759-767. 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**, 333-342. MR**1094928 (92d:33007)****9.**Laforgia, A. and Natalini, P., 2006, On some Turán-type inequalities.*Journal of Inequalities and Applications,***2006**, Article 29828, 6 pp. MR**2221215 (2007c:26028)****10.**Lorch, L., 1994, Monotonicity of the zeros of a cross product of Bessel functions.*Methods and Applications of Analysis*,**1**(1), 75-80. MR**1260384 (95b:33008)****11.**Miller, K.S. and Samko, G., 2001, Completely monotonic functions.*Integral Transforms and Special Functions,***12**, 389-402. MR**1872377 (2002j:26006)****12.**Nåsell, I., 1978, Rational bounds for ratios of modified Bessel functions.*SIAM Journal on Mathematical Analysis,***9**, 1-11. MR**0466662 (57:6539)****13.**Penfold, R., Vanden-Broeck, J.-M. and Grandison, S., 2007, Monotonicity of some modified Bessel function products.*Integral Transforms and Special Functions,***18**(2), 139-144. MR**2290352 (2007m:33015)****14.**Phillips, R.S. and Malin, H., 1950, Bessel function approximations.*American Journal of Mathematics,***72**, 407-418. MR**0035346 (11:720d)****15.**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**, 373-384. MR**0048635 (14:44b)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
33C10,
33C15

Retrieve articles in all journals with MSC (2000): 33C10, 33C15

Additional Information

**Árpád Baricz**

Affiliation:
Faculty of Economics, Babeş-Bolyai University, RO-400591 Cluj-Napoca, Romania

Email:
bariczocsi@yahoo.com

DOI:
https://doi.org/10.1090/S0002-9939-08-09571-3

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