Bohr radius for the bidisc

Description of constant

Let

$${\mathbb{D}}^d:=\{z=(z_1,\dots ,z_d)\in {\mathbb{C}}^d:|z_1|,\dots ,|z_d|<1\}$$

be the unit polydisc, and let the Schur class $S_d$ be the set of analytic functions $f:{\mathbb{D}}^d\to \mathbb{D}$. <a href="#Kne2025-def-polydisc">[Kne2025-def-polydisc]</a> <a href="#Kne2025-def-Schur">[Kne2025-def-Schur]</a>

Writing the power series expansion $f(z)={\sum }_{\alpha \in {\mathbb{N}}_0^d}{f}_{\alpha }{z}^{\alpha }$, define the coefficient-wise ${\ell }^1$ norm $\Vert f{\Vert }_1:={\sum }_{\alpha }|f_{\alpha }|$ and the dilation $f_r(z):=f(rz)$. <a href="#Kne2025-def-l1">[Kne2025-def-l1]</a> <a href="#Kne2025-def-fr">[Kne2025-def-fr]</a>

The Bohr radius $K_d$ is defined by

$$K_d:=\sup \{r>0:\Vert f_r{\Vert }_1\le 1\text{for all }f\in S_d\}.$$

<a href="#Kne2025-def-Kd">[Kne2025-def-Kd]</a>

Equivalently, $K_d$ is the largest number such that for every power series ${\sum }_{\alpha }{c}_{\alpha }{z}^{\alpha }$ with $\left|{\sum }_{\alpha }{c}_{\alpha }{z}^{\alpha }\right|<1$ on ${\mathbb{D}}^d$, one has ${\sum }_{\alpha }|c_{\alpha }{z}^{\alpha }|<1$ whenever ${\max }_{1\le j\le d}|z_j|<K_d$. <a href="#BK1997-def-Kn">[BK1997-def-Kn]</a>

We define

$$C_{59}:=K_2,$$

the Bohr radius for the bidisc ${\mathbb{D}}^2$.

Bohr’s one-variable theorem gives $K_1=1/3$, and in particular implies $K_2\le 1/3$. <a href="#BK1997-Bohr-1d">[BK1997-Bohr-1d]</a> <a href="#BK1997-ub-1-3">[BK1997-ub-1-3]</a>

The exact value of $K_d$ is unknown for every $d>1$; in particular, the exact value of $K_2$ is open. <a href="#BK1997-open">[BK1997-open]</a>

The best established range currently is

$$0.3006\le K_2<\mathrm{0.302825279492.}$$

<a href="#Kne2025-lb-K2-0-3006">[Kne2025-lb-K2-0-3006]</a> <a href="#P2026-ub-K2-0-302825279492">[P2026-ub-K2-0-302825279492]</a>

Known upper bounds

Known lower bounds

Additional comments and links

• Asymptotics in high dimension. The Bohr radius satisfies $K_d\asymp \sqrt{(\log d)/d}$ as $d\to \infty$ (up to absolute constants). <a href="#Kne2025-asymp-Kd">[Kne2025-asymp-Kd]</a>

References

• <a id="BK1997"></a>[BK1997] Boas, Harold P.; Khavinson, Dmitry. Bohr’s power series theorem in several variables. Proceedings of the American Mathematical Society 125 (1997), no. 10, 2975–2979. DOI: https://doi.org/10.1090/S0002-9939-97-04270-6. arXiv PDF: https://arxiv.org/pdf/math/9606203. Google Scholar
• <a id="BK1997-Bohr-1d"></a>[BK1997-Bohr-1d] loc: arXiv v1 PDF p.1, Theorem 1 quote: “Then ${\sum }_{k=0}^{\infty }|c_k{z}^k|<1$ when $|z|<1/3$. Moreover, the radius $1/3$ is the best possible.”
• <a id="BK1997-def-Kn"></a>[BK1997-def-Kn] loc: arXiv v1 PDF p.1, definition paragraph for $K_n$ quote: “Let $K_n$ denote the n-dimensional Bohr radius: the largest number such that if ${\sum }_{\alpha }{c}_{\alpha }{z}^{\alpha }$ converges in the unit polydisc $(z_1,\dots ,z_n):{\max }_{1\le j\le n}|z_j|<1$, and if $\left|{\sum }_{\alpha }{c}_{\alpha }{z}^{\alpha }\right|<1$ in the unit polydisc, then ${\sum }_{\alpha }|c_{\alpha }{z}^{\alpha }|<1$ when ${\max }_{1\le j\le n}|z_j|<K_n$.”
• <a id="BK1997-ub-1-3"></a>[BK1997-ub-1-3] loc: arXiv v1 PDF p.2, paragraph after definition of $K_n$ quote: “It is evident from Bohr’s one-dimensional result that $K_n\le 1/3$ for every $n$.”
• <a id="BK1997-lb-1-3sqrt"></a>[BK1997-lb-1-3sqrt] loc: arXiv v1 PDF p.2, Proof of Theorem 2 quote: “This ball evidently contains the polydisc $z:{\max }_{1\le j\le n}|z_j|<1/(3\sqrt{n})$, whence $K_n\ge 1/(3\sqrt{n})$.”
• <a id="BK1997-open"></a>[BK1997-open] loc: arXiv v1 PDF p.2, Open question quote: “Open question. What is the exact value of the Bohr radius $K_n$ when $n>1$?”

• <a id="Kne2025"></a>[Kne2025] Knese, Greg. Three radii associated to Schur functions on the polydisk. Proceedings of the American Mathematical Society, Series B 12 (2025), no. 5, 48–63. DOI: https://doi.org/10.1090/bproc/262. arXiv PDF: https://arxiv.org/pdf/2410.21693. Google Scholar
• <a id="Kne2025-def-polydisc"></a>[Kne2025-def-polydisc] loc: arXiv v3 PDF p.1, Introduction quote: “$$\begin{gathered} {\mathbb{D}}^d= \\ z=(z_1,\dots ,z_d)\in {\mathbb{C}}^d:|z_1|,\dots ,|z_d|<1 \end{gathered}$$.”
• <a id="Kne2025-def-Schur"></a>[Kne2025-def-Schur] loc: arXiv v3 PDF p.1, Introduction quote: “The Schur class $S_d$ of the polydisk ${\mathbb{D}}^d$ is the set of all analytic $f:{\mathbb{D}}^d\to \mathbb{D}$.”
• <a id="Kne2025-def-l1"></a>[Kne2025-def-l1] loc: arXiv v3 PDF p.1, equation (1.1) context in Introduction quote: “Define the coefficient-wise ${\ell }^1$ norm $\Vert f{\Vert }_1:={\sum }_{\alpha }|f_{\alpha }|$.”
• <a id="Kne2025-def-fr"></a>[Kne2025-def-fr] loc: arXiv v3 PDF p.1, Introduction quote: “For $r>0$ define $f_r(z):=f(rz)$.”
• <a id="Kne2025-def-Kd"></a>[Kne2025-def-Kd] loc: arXiv v3 PDF p.2, Introduction quote: “Define the Bohr radius $K_d$ by $$\begin{gathered} K_d:=\sup \\ r>0:\Vert f_r{\Vert }_1\le 1\text{for all }f\in S_d \end{gathered}$$.”
• <a id="Kne2025-lb-K2-0-3006"></a>[Kne2025-lb-K2-0-3006] loc: arXiv v3 PDF, Corollary 1.2 quote: “Corollary 1.2. $K({\mathcal{A}}_2)=K_2\ge 0.3006$.”
• <a id="Kne2025-asymp-Kd"></a>[Kne2025-asymp-Kd] loc: arXiv v3 PDF p.2, Introduction quote: “After the culmination of deep work by many authors the precise asymptotic $K_d\sim \sqrt{\log d/d}$ was established; see [18], [8].”

• <a id="BPWW2026"></a>[BPWW2026] Baran, Radomił; Pikul, Piotr; Woerdeman, Hugo J.; Wojtylak, Michał. Contractive realization theory for the annulus and other intersections of disks on the Riemann sphere. Journal of Functional Analysis 290 (2026), no. 8, 111346. DOI: https://doi.org/10.1016/j.jfa.2026.111346. arXiv PDF: https://arxiv.org/pdf/2504.03236. Google Scholar
• <a id="BPWW2026-known-interval"></a>[BPWW2026-known-interval] loc: arXiv v1 PDF p.2, Introduction quote: “The constant $K_2$ is the 2-variate version of the Bohr constant, and is known to lie in the interval $(0.3006,1/3)$. We are able to narrow the interval to $(0.3006,0.3177)$ in Theorem 6.4.”
• <a id="BPWW2026-ub-K2-0-3177"></a>[BPWW2026-ub-K2-0-3177] loc: arXiv v1 PDF p.18, Theorem 6.4 quote: “Theorem 6.4. $K_2<0.3177$.”

• <a id="G2026"></a>[G2026] Griego, Sebastian. Rational-inner Fejer averaging certificate for the bidisc Bohr radius bound $K_2<0.3174541$, submitted to this repository (2026).
• <a id="G2026-ub-K2-0-3174541"></a>[G2026-ub-K2-0-3174541] loc: pull request certificate and exact verifier quote: “The exact integer verifier proves $B_{3174541/10000000}(p)>1+1/7307638490$ and the rational-inner Fejer averaging certificate proves $|p|\le 1$ on the bidisc.”

• <a id="P2026"></a>[P2026] Patel, Shivam. Complex phases and a certified upper bound for the bidisc Bohr radius. Research note and reproducibility record, 26 August 2026. MathDB solution. Exact Python and Lean proof package, v1.0.0, Zenodo, 5 September 2026, DOI: 10.5281/zenodo.22341928; GitHub release; pinned source.
• <a id="P2026-ub-K2-0-302825279492"></a>[P2026-ub-K2-0-302825279492] loc: the analytic and finite coefficient certificates in the package README; certificate/EndToEnd.lean, theorem Optim.BohrRadius.bohrRadius_lt_302825279492_div_10pow12 statement: The explicit Schur witness and exact finite coefficient comparison prove $K_2<302825279492/10^{12}=0.302825279492$. The exact value of $K_2$ remains open.

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.