Landau's constant

Description of constant

Let $$\begin{gathered} \mathbb{D}= \\ z\in \mathbb{C}:|z|<1 \end{gathered}$$ and let $\mathcal{F}$ be the class of holomorphic functions $f:\mathbb{D}\to \mathbb{C}$ normalized by $f^{\prime }(0)=1$. <a href="#BS2023-def-F">[BS2023-def-F]</a>

For $f\in \mathcal{F}$, let $L_f$ denote the radius of the largest disk contained in $f(\mathbb{D})$. <a href="#BS2023-def-Lf">[BS2023-def-Lf]</a>

The Landau constant is defined by $$L:=\underset{f\in \mathcal{F}}{\inf }{L}_f.$$ <a href="#BS2023-def-L">[BS2023-def-L]</a>

We define $$C_{57b}:=L.$$

The best recorded bounds in the literature cited below are $$\frac{1}{2}+10^{-335}<L\le \frac{\Gamma (1/3)\Gamma (5/6)}{\Gamma (1/6)}\approx \mathrm{0.5433.}$$ <a href="#BS2023-bounds-L">[BS2023-bounds-L]</a>

Moreover, Rademacher conjectured that the stated upper bound is the exact value of $L$. <a href="#BS2023-rad-conj-L">[BS2023-rad-conj-L]</a>

Known upper bounds

Known lower bounds

Additional comments and links

• 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 $$B\le B_l\le L\le B_u.$$ <a href="#BS2023-relations">[BS2023-relations]</a>

• Wikipedia section on Landau's constant#Bloch's_and_Landau's_constants)

References

• <a id="BS2023"></a>[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
• <a id="BS2023-def-F"></a>[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 $f^{\prime }(0)=1$."
• <a id="BS2023-def-Lf"></a>[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 $f(\mathbb{D})$."
• <a id="BS2023-def-L"></a>[BS2023-def-L] loc: BS2023 PDF p.1269, §1 "Introduction" quote: "$$\begin{gathered} L:=\inf \\ {L}_f:f\in \mathcal{F} \end{gathered}$$."
• <a id="BS2023-bounds-L"></a>[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 $\frac{1}{2}+10^{-335}<L\le \frac{\Gamma (1/3)\Gamma (5/6)}{\Gamma (1/6)}\approx 0.5433$."
• <a id="BS2023-rad-conj-L"></a>[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."
• <a id="BS2023-relations"></a>[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 $B\le B_l\le L\le B_u$."

• <a id="Rad1943"></a>[Rad1943] Rademacher, Hans. On the Bloch-Landau constant. American Journal of Mathematics 65 (1943), no. 3, 387–390. DOI: 10.2307/2371963. Google Scholar

• <a id="Yan1995"></a>[Yan1995] Yanagihara, H. On the locally univalent Bloch constant. Journal d'Analyse Mathématique 65 (1995), 1–17. DOI: 10.1007/BF02788763. Google Scholar

What counts as progress

• A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
• A formal proof (Lean) of a known bound, or a precise error in a claimed one.
• New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.