Functional inequalities for modified Bessel functions
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 of inequalities holds: \[ \begin{multlined} \frac{2K_{\nu}(u_1)K_{\nu}(u_2)}{K_{\nu}(u_1)+K_{\nu}(u_2)}\leq K_{\nu}\Big(\frac{u_1+u_2}{2}\Big)\\ \leq \sqrt{K_{\nu}(u_1)K_{\nu}(u_2)}\leq K_{\nu}(\sqrt{u_1u_2})\leq \frac{K_{\nu}(u_1)+K_{\nu}(u_2)}{2}. \end{multlined} \] Furthermore, the convexity of modified Bessel functions with respect to power mean is investigated. Application to the log-concavity of the gamma-gamma distribution is then presented together with some open problems.
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