  An intriguing correspondence between ingredients in geometric function theory
related to the famous Bieberbach conjecture (de Branges' theorem) and the
non-perturbative crossing symmetric representation of 2-2 scattering amplitudes
of identical scalars is pointed out. Using the dispersion relation and
unitarity, we are able to derive several inequalities, analogous to those which
arise in the discussions of the Bieberbach conjecture. We derive new and strong
bounds on the ratio of certain Wilson coefficients and demonstrate that these
are obeyed in one-loop $\phi^4$ theory, tree level string theory as well as in
the S-matrix bootstrap. Further, we find two sided bounds on the magnitude of
the scattering amplitude, which are shown to be respected in all the contexts
mentioned above. Translated to the usual Mandelstam variables, for large $|s|$,
fixed $t$, the upper bound reads $|\mathcal{M}(s,t)|\lesssim |s^2|$. We discuss
how Szeg\"{o}'s theorem corresponds to a check of univalence in an EFT
expansion, while how the Grunsky inequalities translate into nontrivial,
nonlinear inequalities on the Wilson coefficients.
