Korenblum’s constant
Description of constant
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
\[0.28185\ <\ \kappa\ <\ 0.6778994.\]Hence the best established range currently is
\[0.28185\ <\ C_{68}\ <\ 0.6778994.\]Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $1$ | Trivial from $\kappa\in(0,1)$. | |
| $0.6778994$ | [Wang2008] | Published numerical upper bound recorded in the survey literature. [CS2015-best-range] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $0$ | Trivial from $\kappa>0$. | |
| $0.28185$ | [Wang2011] | Published numerical lower bound recorded in the survey literature. [CS2015-best-range] |
Additional comments and links
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Extremal reformulation. Korenblum’s constant can be expressed via an extremal problem for Bergman-space functions (see the discussion in the survey literature). [CS2015-def-kappa]
References
- [CS2015] Chakraborty, S.; Solynin, A. Korenblum-Type Extremal Problems in Bergman Spaces. (2015). arXiv PDF: https://arxiv.org/pdf/1507.06356.pdf. Publisher: https://arxiv.org/abs/1507.06356. Google Scholar
- [CS2015-def-kappa] loc: arXiv PDF p.1, Abstract quote: “for $f,\ g\in A^2(D)$, there is a constant $c$, $0<c<1$ such that if $|f(z)|\le|g(z)|$ for all $z$ such that $c<|z|<1$, then $\lVert f\rVert_2\le\lVert g\rVert_2$.”
- [CS2015-best-range] loc: arXiv PDF p.2, Section 1 (History and recent results) quote: “In recent papers of Wang [26, 24] the best known bounds to date can be found which are, $0.28185<\kappa<0.6778994$.”
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[Wang2008] Wang, Chunjie. Domination in the Bergman space and Korenblum’s constant. Integral Equations and Operator Theory 61 (2008), 423-432. DOI: https://doi.org/10.1007/s00020-008-1587-4. Google Scholar
- [Wang2011] Wang, Chunjie. Some results on Korenblum’s maximum principle. Journal of Mathematical Analysis and Applications 373 (2011), 393-398. DOI: https://doi.org/10.1016/j.jmaa.2010.07.052. Google Scholar
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.