## On a product of modified Bessel functions

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- by Árpád Baricz
- Proc. Amer. Math. Soc.
**137**(2009), 189-193 - DOI: https://doi.org/10.1090/S0002-9939-08-09571-3
- Published electronically: August 1, 2008
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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.## References

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## Bibliographic Information

**Árpád Baricz**- Affiliation: Faculty of Economics, Babeş-Bolyai University, RO-400591 Cluj-Napoca, Romania
- MR Author ID: 729952
- Email: bariczocsi@yahoo.com
- 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
- Communicated by: Carmen C. Chicone
- © Copyright 2008 American Mathematical Society
- 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
- MathSciNet review: 2439440

Dedicated: Dedicated to my son Koppány