Landau’s constant
Description of constant
Let $\mathbb{D}={z\in\mathbb{C}:\lvert z\rvert<1}$ and let $\mathcal{F}$ be the class of holomorphic functions $f:\mathbb{D}\to\mathbb{C}$ normalized by $f’(0)=1$. [BS2023-def-F]
For $f\in\mathcal{F}$, let $L_f$ denote the radius of the largest disk contained in $f(\mathbb{D})$. [BS2023-def-Lf]
The Landau constant is defined by \(L\ :=\ \inf_{f\in\mathcal{F}} L_f.\) [BS2023-def-L]
We define \(C_{57b}\ :=\ L.\)
The best recorded bounds in the literature cited below are \(\frac{1}{2}+10^{-335}\ <\ L\ \le\ \frac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\ \approx\ 0.5433.\) [BS2023-bounds-L]
Moreover, Rademacher conjectured that the stated upper bound is the exact value of $L$. [BS2023-rad-conj-L]
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $\dfrac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5433$ | [Rad1943] | Upper bound attributed to Rademacher (as summarized in [BS2023]). [BS2023-bounds-L] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $\dfrac{1}{2}+10^{-335}$ | [Yan1995] | Lower bound attributed to Yanagihara (as summarized in [BS2023]). [BS2023-bounds-L] |
Additional comments and links
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Relations to nearby constants. If $B$ is the Bloch constant (entry 57a), $B_l$ the locally univalent Bloch constant, and $B_u$ the univalent Bloch constant (entry 57c), then \(B\ \le\ B_l\ \le\ L\ \le\ B_u.\) [BS2023-relations]
References
- [BS2023] Bhowmik, Bappaditya; Sen, Sambhunath. Improved Bloch and Landau constants for meromorphic functions. Canadian Mathematical Bulletin 66 (2023), 1269–1273. DOI: 10.4153/S0008439523000346. PDF: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/FD465D1F2CEF7E8C62AFF16C3E89B7B4/S0008439523000346a.pdf/improved_bloch_and_landau_constants_for_meromorphic_functions.pdf. Google Scholar
- [BS2023-def-F] loc: BS2023 PDF p.1269, §1 “Introduction” quote: “let $\mathcal{F}$ be the set of all holomorphic functions from $\mathbb{D}$ to the complex plane $\mathbb{C}$ with $f’(0)=1$.”
- [BS2023-def-Lf] loc: BS2023 PDF p.1269, §1 “Introduction” quote: “Given a function $f \in \mathcal{F}$, let $B_f$ be the radius of the largest univalent disk in $f(\mathbb{D})$, and let $L_f$ be the radius of the largest disk in $f(\mathbb{D})$.”
- [BS2023-def-L] loc: BS2023 PDF p.1269, §1 “Introduction” quote: “$L := \inf{L_f : f \in \mathcal{F}}$.”
- [BS2023-bounds-L] loc: BS2023 PDF p.1270, §1 “Introduction” quote: “Rademacher (compare [10]) and Yanagihara (in 1995, see [12]) proved that the upper and the lower bounds for the Landau constant are $\frac{1}{2}+10^{-335}<L\le \frac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5433$.”
- [BS2023-rad-conj-L] loc: BS2023 PDF p.1270, §1 “Introduction” quote: “Rademacher (compare [10]) also conjectured that this upper bound is the precise value of the Landau constant.”
- [BS2023-relations] loc: BS2023 PDF p.1270, §1 “Introduction” quote: “The relation between Bloch constant, Landau constant, locally univalent Bloch constant, and univalent Bloch constant is $B\le B_l\le L\le B_u$.”
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[Rad1943] Rademacher, Hans. On the Bloch-Landau constant. American Journal of Mathematics 65 (1943), no. 3, 387–390. DOI: 10.2307/2371963. Google Scholar
- [Yan1995] Yanagihara, H. On the locally univalent Bloch constant. Journal d’Analyse Mathématique 65 (1995), 1–17. DOI: 10.1007/BF02788763. Google Scholar
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.