  The Bergman $p$-analytic content ($1\leq p<\infty $) of a planar domain
$\Omega $ measures the $L^{p}(\Omega )$-distance between $\overline{z}$ and the
Bergman space $A^{p}(\Omega )$ of holomorphic functions. It has a natural
analogue in all dimensions which is formulated in terms of harmonic vector
fields. This paper investigates isoperimetric inequalities for Bergman
$p$-analytic content in terms of the St Venant functional for torsional
rigidity, and addresses the cases of equality with the upper and lower bounds.
