High order monotonicity of a ratio of the modified Bessel function with applications

Year 2024, Volume: 53 Issue: 6, 1659 - 1673, 28.12.2024

Zhen Hang Yang , Jing-feng Tian

https://doi.org/10.15672/hujms.1244462

Abstract

Let $K_{\mathcal{\nu }}$ be the modified Bessel functions of the second kind of order $\mathcal{\nu }$ and $Q_{\nu }\left( x\right) =xK_{\mathcal{\nu -}1}\left( x\right) /K_{\mathcal{\nu }}\left( x\right) $. In this paper, we proved that $Q_{\mathcal{\nu }}^{\prime \prime \prime }\left( x\right) <\left( >\right) 0$ for $x>0$ if $\left\vert \nu \right\vert >\left( <\right) 1/2$, which gives an affirmative answer to a guess. As applications, some monotonicity and concavity or convexity results as well functional inequalities involving $Q_{\nu }\left( x\right) $ are established. Moreover, several high order monotonicity of $x^{k}Q_{\nu }^{\left( n\right) }\left( x\right) $ on $\left( 0,\infty \right) $ for certain integers $k$ and $n$ are given.

Keywords

modified Bessel function of the second kind, high order monotonicity, complete monotonicity, functional inequality

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