  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.
