Korenblum-Type Extremal Problems in Bergman Spaces

TL;DR

This paper investigates extremal problems in Bergman spaces related to Korenblum's conjecture, establishing existence, boundedness, and properties of solutions, and discussing bounds and special cases.

Contribution

It proves the existence and boundedness of extremal functions in Bergman spaces for Korenblum's conjecture and explores properties and bounds of solutions.

Findings

Existence of solutions in Bergman spaces established.

Extremal functions are proven to be bounded.

Discussion of bounds and special polynomial cases.

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.