  We explore consequences of the Averaged Null Energy Condition (ANEC) for
scaling dimensions $\Delta$ of operators in four-dimensional $\mathcal{N}=1$
superconformal field theories. We show that in many cases the ANEC bounds are
stronger than the corresponding unitarity bounds on $\Delta$. We analyze in
detail chiral operators in the $(\frac12 j,0)$ Lorentz representation and prove
that the ANEC implies the lower bound $\Delta\ge\frac32j$, which is stronger
than the corresponding unitarity bound for $j>1$. We also derive ANEC bounds on
$(\frac12 j,0)$ operators obeying other possible shortening conditions, as well
as general $(\frac12 j,0)$ operators not obeying any shortening condition. In
both cases we find that they are typically stronger than the corresponding
unitarity bounds. Finally, we elucidate operator-dimension constraints that
follow from our $\mathcal{N}=1$ results for multiplets of $\mathcal{N}=2,4$
superconformal theories in four dimensions. By recasting the ANEC as a convex
optimization problem and using standard semidefinite programming methods we are
able to improve on previous analyses in the literature pertaining to the
nonsupersymmetric case.
