  Parabolic Cylinder functions (PCFs) are classical special functions with
applications in many different fields. However, there is little information
available regarding simple uniform approximations and bounds for these
functions. We obtain very sharp bounds for the ratio
$\Phi_n(x)=U(n-1,x)/U(n,x)$ and the double ratio $\Phi_n(x)/\Phi_{n+1}(x)$ in
terms of elementary functions (algebraic or trigonometric) and prove the
monotonicity of these ratios; bounds for $U(n,z)/U(n,y)$ are also made
available. The bounds are very sharp as $x\rightarrow \pm \infty$ and
$n\rightarrow +\infty$, and this simultaneous sharpness in three different
directions explains their remarkable global accuracy. Upper and lower
elementary bounds are obtained which are able to produce several digits of
accuracy for moderately large $|x|$ and/or $n$.
