Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind

Zhong-Xuan Mao; Jing-Feng Tian

doi:10.5802/crmath.399

Partial differential equations

Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind

Zhong-Xuan Mao ¹; Jing-Feng Tian ²

¹ Department of Mathematics and Physics, North China Electric Power University,Yonghua Street 619, 071003, Baoding, P. R. China
² Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, 071003, Baoding, P. R. China

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 217-235.

Abstract

In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function $- T_{ν, α, β} (s)$ is completely monotonic in $s$ and absolutely monotonic in $ν$ if and only if $β \geq 1$ , where $T_{ν, α, β} (s) = K_{ν}^{2} (s) - β K_{ν - α} (s) K_{ν + α} (s)$ defined on $s > 0$ and $K_{ν} (s)$ is the modified Bessel function of the second kind of order $ν$ . Finally, we determine the necessary and sufficient conditions for the functions $s \mapsto T_{μ, α, 1} (s) / T_{ν, α, 1} (s)$ , $s \mapsto (T_{μ, α, 1} (s) + T_{ν, α, 1} (s)) / (2 T_{(μ + ν) / 2, α, 1} (s))$ , and $s \mapsto \frac{d^{n_{1}}}{d ν^{n_{1}}} T_{ν, α, 1} (s) / \frac{d^{n_{2}}}{d ν^{n_{2}}} T_{ν, α, 1} (s)$ to be monotonic in $s \in (0, \infty)$ by employing the monotonicity rules.

Received: 2022-03-14
Accepted: 2022-06-18
Published online: 2023-01-12

DOI: 10.5802/crmath.399

Classification: 33C10, 33B15, 26A48

Author's affiliations:

Zhong-Xuan Mao ¹; Jing-Feng Tian ²

¹ Department of Mathematics and Physics, North China Electric Power University,Yonghua Street 619, 071003, Baoding, P. R. China
² Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, 071003, Baoding, P. R. China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Zhong-Xuan Mao and Jing-Feng Tian},
     title = {Monotonicity and complete monotonicity of some functions involving the modified {Bessel} functions of the second kind},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {217--235},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.399},
     language = {en},
}

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Zhong-Xuan Mao; Jing-Feng Tian. Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 217-235. doi : 10.5802/crmath.399. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.399/

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