A maximum principle for the Bergman space (Q1182640)

Let \(f\), \(g\) be analytic in the unit disk \(\mathbb{D}\), and let \(f\) have a zero of order \(\geq k\) in \(\zeta\in\mathbb{D}\) if \(g\) has a zero of order \(k\) in \(\zeta\). The author proves, that \(| f(z)|\geq| g(z)|\) for \({1\over2}e^{-2}<| z|<1\) implies the inequality \[ \int_ \mathbb{D} | g(z)|^ 2 dm\leq\int_ \mathbb{D} | f(z)|^ 2 dm, \] where \(m\) is the Lebesgue measure.