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 $Deltagefrac32j$, 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.
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