  For a rational map $f$ and a H\"older continuous potential $\phi$ satisfying
$$ \sup\phi< P(f,\phi)$$ the uniqueness and stochastic properties of the
corresponding equilibrium states have been extensively studied. For a given
rational map $f$, in this paper we characterize those H\"older continuous
potentials $\phi$ for which this property is satisfied for some iterate of $f$.
