Modified Bessel function of the second kind high order monotonicity complete monotonicity functional inequality
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.
Modified Bessel function of the second kind high order monotonicity complete monotonicity functional inequality 2000 Mathematics Subject Classification. 33C10, 26A51
Birincil Dil | İngilizce |
---|---|
Konular | Matematik |
Bölüm | Matematik |
Yazarlar | |
Erken Görünüm Tarihi | 14 Nisan 2024 |
Yayımlanma Tarihi | |
Yayımlandığı Sayı | Yıl 2024 Erken Görünüm |