Functional inequalities for modified Bessel functions
type: Item
instance of: scholarly article
title: Functional inequalities for modified Bessel functions
zbMATH Open document ID: 1237.39023
author: Árpád Baricz
author: Saminathan Ponnusamy
author: M. Vuorinen
published in: Expositiones Mathematicae
publication date: 2011-12-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.4814
review text: Denote by \(I_{\nu}\) and \(K_{\nu}\) the modified Bessel functions of the first and second kind of real order \(\nu\). The authors prove the following results: Theorem 1. The following assertions are true: {\parindent=7mm \begin{itemize}\item[(a)]\(u \mapsto uI'_{\nu}/I^2_{\nu}\) is strictly decreasing on \((0,\infty)\) for all \(\nu \geq 1\); \item[(b)]\(u \mapsto uI_{\nu}/I^2_{\nu}\) is strictly increasing on \((0,\infty)\) for all \(\nu > -1\); \item[(c)]\(u \mapsto \sqrt{u}I_{\nu}\) is strictly log-concave on \((0,\infty)\) for all \(\nu \geq 1/2\); \item[(d)]\(u \mapsto u^2I'_{\nu}/I^2_{\nu}\) is strictly increasing on \((0,\infty)\) for all \(\nu \geq \nu_0\), where \(\nu_0\) is the positive root of the cubic equation \(8\nu^3-9\nu^2-2\nu-1=0\). \end{itemize}} In particular, for all \(u_1, u_2>0\) and \(\nu \geq \nu_0\) the following chain of inequalities holds: \[ \begin{multlined} \frac{2I_{\nu}(u_1)I_{\nu}(u_2)}{I_{\nu}(u_1)+I_{\nu}(u_2)}\leq I_{\nu}\Big(\frac{2u_1u_2}{u_1+u_2}\Big)\\ \leq I_{\nu}(\sqrt{u_1u_2}) \leq \sqrt{I_{\nu}(u_1)I_{\nu}(u_2)}\leq \sqrt{\frac{u_1+u_2}{2\sqrt{u_1u_2}}}\cdot I_{\nu}\Big(\frac{u_1+u_2}{2}\Big).\end{multlined} \] Theorem 2. The following assertions are true: {\parindent=7mm \begin{itemize}\item[(a)]\(u \mapsto K'_{\nu}/K^2_{\nu}\) is strictly decreasing on \((0,\infty)\) for all \(|\nu| \geq 1\); \item[(b)]\(u \mapsto K'_{\nu}/K_{\nu}\) is strictly increasing on \((0,\infty)\) for all \(\nu\in \mathbb R\); \item[(c)]\(u \mapsto uK'_{\nu}/K_{\nu}\) is strictly decreasing on \((0,\infty)\) for all \(\nu\in \mathbb R\); \item[(d)]\(u \mapsto uK'_{\nu}\) is strictly increasing on \((0,\infty)\) for all \(\nu\in \mathbb R\); \item[(e)]\(u \mapsto u^2K'_{\nu}\) is strictly increasing on \((0,\infty)\) for all \(|\nu|\geq 5/4\); \item[(f)]\(u \mapsto u^2K'_{\nu}\) is strictly increasing on \((2,\infty)\) for all \(\nu \in \mathbb R\). \end{itemize}} In particular, for all \(u_1, u_2>0\) and \(|\nu| \geq 1\) the following chain o…
reviewed by: Gian Luigi Forti
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 5992157
zbMATH Keywords: functional inequalities
zbMATH Keywords: modified Bessel functions
zbMATH Keywords: convexity
zbMATH Keywords: log-convexity
zbMATH Keywords: geometrical convexity
zbMATH Keywords: gamma-gamma distribution
zbMATH Keywords: Turán-type inequality
MaRDI profile type: Publication
OpenAlex ID: W3102288529
arXiv ID: 1009.4814
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DOI: 10.1016/J.EXMATH.2011.07.001
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description: scientific article