On a maximum principle in Bergman space (Q5943739)

The following theorem is proved. Let \(f\) and \(g\) be analytic functions in the unit disk \(\mathbb{D}\) such that \(|f(z)|\leq|g(z) |\) in \(\{c< |z|<1\}\), where \(c=0.15724\), then for all \(p\geq 1\), \[ \int_\mathbb{D} |f|^p dm\leq\int_\mathbb{D} |g|^p dm. \] A weaker result (with \(p=2\) and \(c={1\over 25})\) was proved by Hayman [\textit{W. K. Hayman}, Analysis 19, 195-205 (1999; Zbl 0934.30020)] as a solution to Korenblum's conjecture of 1991. The method uses Lehto's generalization of the Schwarz lemma, and the specific value of the constant \(c\) comes from numerical calculations.