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on Korenblum's constant ($C_{68}$)

The statement

Let $\mathbb{D}:=\{z\in\mathbb{C}:\lvert z\rvert<1\}$ . The Bergman space $\mathcal{A}^2(\mathbb{D})$ consists of analytic functions $f$ on $\mathbb{D}$ with

\lVert f\rVert_2\ :=\ \left(\frac{1}{\pi}\int_{\mathbb{D}} \lvert f(z)\rvert^2\,dA(z)\right)^{1/2}\ <\ \infty,

where $dA(z)$ denotes the Lebesgue area measure.

For $c\in(0,1)$ , write

A(c,1)\ :=\ \{z\in\mathbb{C}:\ c<\lvert z\rvert<1\}.

Korenblum's maximum principle asserts that there exists a constant $\kappa\in(0,1)$ such that whenever $f,g$ are analytic in $\mathbb{D}$ and

\lvert f(z)\rvert\ \le\ \lvert g(z)\rvert \qquad (z\in A(\kappa,1)),

one has $\lVert f\rVert_2\le \lVert g\rVert_2$ . [CS2015-def-kappa]

We define

C_{68}\ :=\ \kappa,

where $\kappa$ is the largest constant for which this implication holds (often called Korenblum's constant). [CS2015-def-kappa]

The cited literature gives

0.3554\ <\ \kappa\ <\ 0.6778994.

[CS2015-best-range] [Wang2025]

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