  We study properties of asymptotically free vectorial gauge theories with
gauge groups $G={\rm SO}(N_c)$ and $G={\rm Sp}(N_c)$ and $N_f$ fermions in a
representation $R$ of $G$, at an infrared (IR) zero of the beta function,
$\alpha_{IR}$, in the non-Abelian Coulomb phase. The fundamental, adjoint, and
rank-2 symmetric and antisymmetric tensor fermion representations are
considered. We present scheme-independent calculations of the anomalous
dimensions of (gauge-invariant) fermion bilinear operators
$\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ and of the derivative of the beta
function at $\alpha_{IR}$, denoted $\beta'_{IR}$, to $O(\Delta_f^5)$, where
$\Delta_f$ is an $N_f$-dependent expansion variable. It is shown that all
coefficients in the expansion of $\gamma_{\bar\psi\psi,IR}$ that we calculate
are positive for all representations considered, so that to $O(\Delta_f^4)$,
$\gamma_{\bar\psi\psi,IR}$ increases monotonically with decreasing $N_f$ in the
non-Abelian Coulomb phase. Using this property, we give a new estimate of the
lower end of this phase for some specific realizations of these theories.
