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Korenblum's constant (C68C_{68})

Korenblum's constant: determine C68C_{68} or improve its bounds; the repository records 0.3554 (lower) and 0.6778994 (upper).

MSC 30H20 · Bergman spaces and Fock spaces

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

Let D:={z∈C:∣z∣<1}\mathbb{D}:=\{z\in\mathbb{C}:\lvert z\rvert<1\}. The Bergman space A2(D)\mathcal{A}^2(\mathbb{D}) consists of analytic functions ff on D\mathbb{D} with

∥f∥2 := (1π∫D∣f(z)∣2 dA(z))1/2 < ∞,\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)dA(z) denotes the Lebesgue area measure.

For c∈(0,1)c\in(0,1), write

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

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

∣f(z)∣ ≤ ∣g(z)∣(z∈A(κ,1)),\lvert f(z)\rvert\ \le\ \lvert g(z)\rvert \qquad (z\in A(\kappa,1)),

one has ∥f∥2≤∥g∥2\lVert f\rVert_2\le \lVert g\rVert_2. [CS2015-def-kappa]

We define

C68 := κ,C_{68}\ :=\ \kappa,

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

The cited literature gives

0.3554 < κ < 0.6778994.0.3554\ <\ \kappa\ <\ 0.6778994.

[CS2015-best-range] [Wang2025]

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Background

References (from the repository page)

  • [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∈A2(D)f,\ g\in A^2(D), there is a constant cc, 0<c<10<c<1 such that if ∣f(z)∣≤∣g(z)∣|f(z)|\le|g(z)| for all zz such that c<∣z∣<1c<|z|<1, then ∥f∥2≤∥g∥2\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<κ<0.67789940.28185<\kappa<0.6778994."

  • [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

  • [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

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0.3554; best known upper bound 0.6778994. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
11 Trivial from κ∈(0,1)\kappa\in(0,1).
0.67789940.6778994 [Wang2008] Published numerical upper bound recorded in the survey literature. [CS2015-best-range]

Known lower bounds

Bound Reference Comments
00 Trivial from κ>0\kappa>0.
0.281850.28185 [Wang2011] Published numerical lower bound recorded in the survey literature. [CS2015-best-range]
0.35540.3554 [Wang2025] Published numerical lower bound.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Korenblum's constant ($C_{68}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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