  We investigate some higher-loop structural properties of the $\beta$ function
in asymptotically free vectorial gauge theories. Our main focus is on theories
with fermion contents that lead to an infrared (IR) zero in $\beta$. We present
analytic and numerical calculations of the value of the gauge coupling where
$\beta$ reaches a minimum, the value of $\beta$ at this minimum, and the slope
of $\beta$ at the IR zero, at two-, three-, and four-loop order. The slope of
$\beta$ at the IR zero is relevant for estimates of a dilaton mass in
quasiconformal gauge theories. Some inequalities are derived concerning the
dependence of the above quantities on loop order. A general inequality is
derived concerning the dependence of the shift of the IR zero of $\beta$, from
the $n$-loop to the $(n+1)$-loop order, on the sign of the $(n+1)$-loop
coefficient in $\beta$. Some results are also given for gauge theories with
${\cal N}=1$ supersymmetry.
