A maximum principle for the Bergman space.
Let f(z) and g(z) be holomorphic in the open unit disk D and let Zf and Zg be their zero sets. If Zf ⊃ Zg and |f(z)| ≥ |g(z)| (1/2 e-2 < |z| < 1), then || f || ≥ || g || where || · || is the Bergman norm: || f ||2 = π-1 ∫D |f(z)|2 dm (dm is the Lebesgue area measure).