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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a product of modified Bessel functions


Author: Árpád Baricz
Journal: Proc. Amer. Math. Soc. 137 (2009), 189-193
MSC (2000): Primary 33C10, 33C15
Published electronically: August 1, 2008
MathSciNet review: 2439440
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Abstract: Let $ I_{\nu}$ and $ K_{\nu}$ denote the modified Bessel functions of the first and second kinds of order $ \nu.$ In this note we prove that the monotonicity of $ u\mapsto I_{\nu}(u)K_{\nu}(u)$ on $ (0,\infty)$ for all $ \nu\geq -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 $ \nu.$ Moreover, we show that the function $ u\mapsto I_{\nu}(u)K_{\nu}(u)$ is strictly completely monotonic on $ (0,\infty)$ for all $ \nu\in[-1/2,1/2].$ At the end of this note, a conjecture is stated.


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Additional Information

Árpád Baricz
Affiliation: Faculty of Economics, Babeş-Bolyai University, RO-400591 Cluj-Napoca, Romania
Email: bariczocsi@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09571-3
PII: S 0002-9939(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