  Dual pairs of interior and exterior Hardy spaces associated to a simple
closed Lipschitz planar curve are considered, leading to a M\"obius invariant
function bounding the norm of the Cauchy transform $\bf{C}$ from below. This
function is shown to satisfy strong rigidity properties and is closely
connected via the Berezin transform to the square of the Kerzman-Stein
operator. Explicit example calculations are presented. For ellipses, a new
asymptotically sharp lower bound on the norm of $\bf{C}$ is produced.
