On Korenblum's maximum principle for the Bergman space (Q1805519)

\textit{B. Korenblum} [Publ. Mat., Barc. 35, No. 2, 479-486 (1991; Zbl 0758.30020)] has conjectured the following maximum principle for the Bergman space: there exists a constant \(c \in (0,1)\) such that if \(f\) and \(g\) are square area-integrable analytic functions on the open unit disc \(\mathbb{D}\), with \(| f(z) | \geq | g(z) |\) on the annulus \(c \leq | z | < 1\) then \(\| f \|_ 2 \geq \| g \|_ 2\). We prove the above conjecture when either \(f\) or \(g\) is a monomial. For \(f(z) = z^ k\) the choice \(c = c_ k = ({1 \over (k + 1) (k + 2)})^{1/2k}\) and for \(g(z) = z^ k\) the choice \(c = d_ k = \sqrt {{k + 1 \over k + 2}}\) is sufficient for Korenblum's conjecture to hold. The constant \(d_ k\) is sharp. Furthermore, Korenblum's conjecture holds for \(g(z) = a_ kz^ k + a_{k+1} z^{k+1}\), with \(c = d_ k\).