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]
Including the unrefereed certificate below, the recorded bounds are
\[0.432^{*}\ \le\ \kappa\ <\ 0.6778994.\][CS2015-best-range] [Wang2025] [Wik2026] [Pin2026]
The asterisk marks a self-archived, unrefereed computer-assisted claim that has not yet been independently replayed by a third party. The strongest lower bound located without that marker is Wikström’s $0.4263$ [Wik2026].
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] |
| $0.3554$ | [Wang2025] | Published numerical lower bound. |
| $0.4263$ | [Wik2026] | Eight rational atoms plus a Lebesgue tail on $[0.99,1]$; Wang’s annular coefficient estimate and Wikström’s moment-duality criterion, with an Arb certificate. Unrefereed arXiv preprint; software at doi:10.5281/zenodo.21446369. |
| $0.432$* | [Pin2026] | $c=54/125$, $R=0.999$, and $d\mu=\sum_{j=1}^{12}w_j\,d\delta_{t_j}+\mathbf1_{[R,1]}dt$, with exact terminating-decimal pairs $(t_j,w_j)$: (0.186624, 0.401004850339305473); (0.587760864611916678, 0.334589004564615711); (0.8229718555012977, 0.152356299682819746); (0.924010443899645262, 0.062924486526793319); (0.965990583618882126, 0.026544759438047238); (0.984044289625064295, 0.011714716766935399); (0.992173624211215355, 0.00540693721877966); (0.995987383331351062, 0.002581897617100729); (0.997819789755709063, 0.001239794794409609); (0.998676970469441039, 0.000541241719003444); (0.998976894402894056, 0.000064073011221399); (0.999, 0.000031938320968273). The weights sum exactly to $R$. Replaces the annular coefficient estimate by a two-zero covering-strip Schwarz bound and classical hyperbolic Jung, retaining Wikström’s moment criterion. A lower bound $1/G(t)\ge(2563746486913/10^{14})(1-t)$ on $[R,1]$ makes the tail integrals elementary. Two separately implemented exact checkers (Arb and standard-library integer intervals) verify first moments through 31278 and second moments through 23999, with analytic tails from 31279 and 24000, respectively; the second tail uses $[0.9998,0.9999]$. Certificate and paper: repository, commit 9d48de64987911f5a07a397323345be8b97770a8. Asterisked: self-archived, unrefereed, no third-party replay or proof-assistant formalization; the checkers verify quantitative hypotheses, and the analytic proof requires mathematical review. |
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
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[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
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[Wang2025] Wang, Chunjie. A lower bound on Korenblum’s constant. Proceedings of the American Mathematical Society 153 (2025), 1209-1214. DOI: https://doi.org/10.1090/proc/17112. Google Scholar
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[Wik2026] Wikström, Frank. Moment duality and an improved lower bound for Korenblum’s constant. Unrefereed preprint (2026). arXiv:2607.17748. Certified lower bound 0.4263; accompanying software.
- [Pin2026] Pinho, Ricardo. A covering-strip and moment certificate for Korenblum’s constant. Unrefereed preprint (19 October 2026). Paper PDF, certificate and independent exact checkers (MIT). ORCID 0009-0006-2744-2599. No optimality claimed.
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.
The [Pin2026] contribution was prepared by AI agents under Ricardo Pinho’s direction, including analysis, proof drafting, certificate, checkers, and submission text. Author statement: “Analysis and proof drafting by AI agents under my direction; I reviewed the results at summary level.” This does not claim a line-by-line human verification; independent review is needed.