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.
