Korenblum-Type Extremal Problems in Bergman Spaces

Abstract

We shall study non-linear extremal problems in Bergman space $\mathcal{A}^2(\mathbb{D})$. We show the existence of the solution and that the extremal functions are bounded. Further, we shall discuss special cases for polynomials, investigate the properties of the solution and provide a bound for the solution. This problem is an equivalent formulation of B. Korenblum's conjecture, also known as Korenblum's Maximum Principle: for $f$, $g\in \mathcal{A}^2(\mathbb{D})$, there is a constant $c$, $0<c<1$ such that if $|f(z)|\leq |g(z)|$ for all $z$ such that $c<|z|<1$, then $\|f\|_2\leq \|g\|_2$. The existence of such $c$ was proved by W. Hayman but the exact value of the best possible value of $c$, denoted by $\kappa$, remains unknown.