  Two elementary and classical results about the Bessel quotient $y_\nu =
\frac{I_{\nu+1}}{I_\nu}$ state that on the half-line $(0,\infty)$ one has for
$\nu\ge -1/2$: \begin{itemize} \item[(i)] $0 < y_\nu< 1$; \item[(ii)] $y_\nu$
is strictly increasing. \end{itemize} In this paper we show that (i) and (ii)
have some nontrivial and interesting applications to pde's. As a consequence of
them, we establish some sharp new results for a class of degenerate partial
differential equations of parabolic type in $\Rnp\times (0,\infty)$ which arise
in connection with the analysis of the fractional heat operator $(\p_t -
\Delta)^s$ in $\Rn\times (0,\infty)$, see Theorems 1.2, 1.4, 1.5 and 1.7 below.
