### Abstract

Let I _{v} and K _{v} denote the modified Bessel functions of the first and second kinds of order v. In this note we prove that the monotonicity of u I _{v}(u)K _{v}(u)on(0, ∞)for all v≥-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 v. Moreover, we show that the function u I _{v}(u)K _{v}(u) is strictly completely monotonic on (0, ∞) for all v ∈ [-1/2,1/2]. At the end of this note, a conjecture is stated.

Original language | English |
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Pages (from-to) | 189-193 |

Number of pages | 5 |

Journal | Proceedings of the American Mathematical Society |

Volume | 137 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2009 |

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### Keywords

- Complete monotonicity
- Modified Bessel functions
- Turán type inequalities

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**On a product of modified Bessel functions.** / Baricz, A.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 137, no. 1, pp. 189-193. https://doi.org/10.1090/S0002-9939-08-09571-3

}

TY - JOUR

T1 - On a product of modified Bessel functions

AU - Baricz, A.

PY - 2009/1

Y1 - 2009/1

N2 - Let I v and K v denote the modified Bessel functions of the first and second kinds of order v. In this note we prove that the monotonicity of u I v(u)K v(u)on(0, ∞)for all v≥-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 v. Moreover, we show that the function u I v(u)K v(u) is strictly completely monotonic on (0, ∞) for all v ∈ [-1/2,1/2]. At the end of this note, a conjecture is stated.

AB - Let I v and K v denote the modified Bessel functions of the first and second kinds of order v. In this note we prove that the monotonicity of u I v(u)K v(u)on(0, ∞)for all v≥-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 v. Moreover, we show that the function u I v(u)K v(u) is strictly completely monotonic on (0, ∞) for all v ∈ [-1/2,1/2]. At the end of this note, a conjecture is stated.

KW - Complete monotonicity

KW - Modified Bessel functions

KW - Turán type inequalities

UR - http://www.scopus.com/inward/record.url?scp=77950583212&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950583212&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-08-09571-3

DO - 10.1090/S0002-9939-08-09571-3

M3 - Article

AN - SCOPUS:77950583212

VL - 137

SP - 189

EP - 193

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 1

ER -