  We apply the average null energy condition to obtain upper bounds on the
three-point function coefficients of stress tensors and a scalar operator,
$\langle TT {\cal O } \rangle,$ 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 $\langle TT J \rangle$. 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 $\int
\phi\hspace{.5mm} 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 $\phi \hspace{.5mm}W W^*$.
