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

Year 2024, Early Access, 1 - 15

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 conjecture. 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 found.

Keywords

Modified Bessel function of the second kind, high order monotonicity, complete monotonicity, functional inequality, 2000 Mathematics Subject Classification. 33C10, 26A51

References

• [1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Dover Publications, New York and Washington, 1972.