  The operator product expansion (OPE) in 4d (super)conformal field theory is
of broad interest, for both formal and phenomenological applications. In this
paper, we use conformal perturbation theory to study the OPE of nearly-free
fields coupled to SCFTs. Under fairly general assumptions, we show that the OPE
of a chiral operator of dimension $\Delta = 1+\epsilon$ with its complex
conjugate always contains an operator of dimension less than $2 \Delta$. Our
bounds apply to Banks-Zaks fixed points and their generalizations, as we
illustrate using several examples.
