  We give new sufficient conditions for a compact set $E \subseteq \mathbb{C}$
to satisfy $\gamma(E)=\gamma_c(E)$, where $\gamma$ is the analytic capacity and
$\gamma_c$ is the Cauchy capacity. As a consequence, we provide examples of
compact plane sets such that the above equality holds but the Ahlfors function
is not the Cauchy transform of any complex Borel measure supported on the set.
