The Bergman p-analytic content (1 <= p < infinity) of a planar domain Omega measures the L-p (Omega)-distance between (z) over bar 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.