Monotonicity properties for ratios and products of modified Bessel functions and sharp trigonometric bounds

Javier Segura

Published 2021-05-06Version 1

Let $I_{\nu}(x)$ and $K_{\nu}(x)$ be the first and second kind modified Bessel functions. It is shown that the nullclines of the Riccati equation satisfied by $x^{\alpha} \Phi_{i,\nu}(x)$, $i=1,2$, with $\Phi_{1,\nu}=I_{\nu-1}(x)/I_{\nu}(x)$ and $\Phi_{2,\nu}(x)=-K_{\nu-1}(x)/K_{\nu}(x)$, are bounds for $x^{\alpha} \Phi_{i,\nu}(x)$, which are solutions with unique monotonicity properties; these bounds hold at least for $\pm \alpha\notin (0,1)$ and $\nu\ge 1/2$. Properties for the product $P_{\nu}(x)=I_{\nu}(x)K_{\nu}(x)$ can be obtained as a consequence; for instance, it is shown that $P_{\nu}(x)$ is decreasing if $\nu\ge -1$ (extending the known range of this result) and that $xP_{\nu}(x)$ is increasing for $\nu\ge 1/2$. We also show that the double ratios $W_{i,\nu}(x)=\Phi_{i,\nu+1}(x)/\Phi_{i,\nu}(x)$ are monotonic and that these monotonicity properties are exclusive of the first and second kind modified Bessel functions. Sharp trigonometric bounds can be extracted from the monotonicity of the double ratios. The trigonometric bounds for the ratios and the product are very accurate as $x\rightarrow 0^+$, $x\rightarrow +\infty$ and $\nu\rightarrow +\infty$ in the sense that the first two terms in the power series expansions in these limits are exact.