| Summary: | Abstract We apply the average null energy condition to obtain upper bounds on the three-point function coefficients of stress tensors and a scalar operator, TTO $$ \left\langle TT\mathcal{O}\right\rangle $$, in general CFTs. We also constrain the gravitational anomaly of U(1) currents in four-dimensional CFTs, which are encoded in three-point functions of the form 〈T T J 〉. In theories with a large N AdS dual we translate these bounds into constraints on the coefficient of a higher derivative bulk term of the form ∫ϕ W 2. We speculate that these bounds also apply in de-Sitter. In this case our results constrain inflationary observables, such as the amplitude for chiral gravity waves that originate from higher derivative terms in the Lagrangian of the form ϕ W W ∗.
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