  We consider an asymptotically free, vectorial, ${\cal N}=1$ supersymmetric
gauge theory with gauge group $G$ and $N_f$ pairs of chiral superfields in the
respective representations ${\cal R}$ and $\bar {\cal R}$ of $G$, having an
infrared fixed point (IRFP) of the renormalization group at $\alpha_{IR}$. We
present exact results for the anomalous dimensions of various (gauge-invariant)
composite chiral superfields $\gamma_{{\Phi}_{\rm prod}}$ at the IRFP and prove
that these increase monotonically with decreasing $N_f$ in the non-Abelian
Coulomb phase of the theory and that scheme-independent expansions for these
anomalous dimensions as powers of an $N_f$-dependent variable, $\Delta_f$,
exhibit monotonic and rapid convergence to the exact $\gamma_{{\Phi}_{\rm
prod}}$ throughout this phase. We also present a scheme-independent calculation
of the derivative of the beta function, $d\beta/d\alpha
|_{\alpha=\alpha_{IR}}$, denoted $\beta'_{IR}$, up to $O(\Delta_f^3)$ for
general $G$ and ${\cal R}$, and, for the case $G={\rm SU}(N_c)$, ${\cal R}=F$,
we give an analysis of the properties of $\beta'_{IR}$ calculated to
$O(\Delta_f^4)$.
