Bohr radius for the bidisc

Description of constant

Let

be the unit polydisc, and let the Schur class $S_d$ be the set of analytic functions $f:\mathbb{D}^d\to\mathbb{D}$. [Kne2025-def-polydisc] [Kne2025-def-Schur]

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 $\lVert f\rVert_1:=\sum_\alpha \lvert f_\alpha\rvert$ and the dilation $f_r(z):=f(rz)$. [Kne2025-def-l1] [Kne2025-def-fr]

The Bohr radius $K_d$ is defined by

[Kne2025-def-Kd]

Equivalently, $K_d$ is the largest number such that for every power series $\sum_\alpha c_\alpha z^\alpha$ with $\bigl\lvert\sum_\alpha c_\alpha z^\alpha\bigr\rvert<1$ on $\mathbb{D}^d$, one has $\sum_\alpha \lvert c_\alpha z^\alpha\rvert<1$ whenever $\max_{1\le j\le d}\lvert z_j\rvert<K_d$. [BK1997-def-Kn]

We define

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

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

The exact value of $K_d$ is unknown for every $d>1$; in particular, the exact value of $K_2$ is open. [BK1997-open]

The best established range currently is

[Kne2025-lb-K2-0-3006] [BPWW2026-ub-K2-0-3177]

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). [Kne2025-asymp-Kd]

References

• [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
  • [BK1997-Bohr-1d] loc: arXiv v1 PDF p.1, Theorem 1 quote: “Then $\sum_{k=0}^\infty \lvert c_k z^k\rvert<1$ when $\lvert z\rvert<1/3$. Moreover, the radius $1/3$ is the best possible.”
  • [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 ${(z_1,\ldots,z_n):\max_{1\le j\le n}\lvert z_j\rvert<1}$, and if $\left\lvert\sum_\alpha c_\alpha z^\alpha\right\rvert<1$ in the unit polydisc, then $\sum_\alpha \lvert c_\alpha z^\alpha\rvert<1$ when $\max_{1\le j\le n}\lvert z_j\rvert<K_n$.”
  • [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 $n$.”
  • [BK1997-lb-1-3sqrt] loc: arXiv v1 PDF p.2, Proof of Theorem 2 quote: “This ball evidently contains the polydisc ${z:\max_{1\le j\le n}\lvert z_j\rvert<1/(3\sqrt{n})}$, whence $K_n\ge 1/(3\sqrt{n})$.”
  • [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 $n>1$?”
• [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
  • [Kne2025-def-polydisc] loc: arXiv v3 PDF p.1, Introduction quote: “$\mathbb{D}^d={z=(z_1,\dots,z_d)\in\mathbb{C}^d:\ \lvert z_1\rvert,\dots,\lvert z_d\rvert<1}$.”
  • [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 $f:\mathbb{D}^d\to\mathbb{D}$.”
  • [Kne2025-def-l1] loc: arXiv v3 PDF p.1, equation (1.1) context in Introduction quote: “Define the coefficient-wise $\ell^1$ norm $\lVert f\rVert_1:=\sum_\alpha \lvert f_\alpha\rvert$.”
  • [Kne2025-def-fr] loc: arXiv v3 PDF p.1, Introduction quote: “For $r>0$ define $f_r(z):=f(rz)$.”
  • [Kne2025-def-Kd] loc: arXiv v3 PDF p.2, Introduction quote: “Define the Bohr radius $K_d$ by $K_d:=\sup{r>0:\ \lVert f_r\rVert_1\le 1\ \text{for all } f\in S_d}$.”
  • [Kne2025-lb-K2-0-3006] loc: arXiv v3 PDF, Corollary 1.2 quote: “Corollary 1.2. $K(\mathcal{A}_2)=K_2\ge 0.3006$.”
  • [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].”
• [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
  • [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.”
  • [BPWW2026-ub-K2-0-3177] loc: arXiv v1 PDF p.18, Theorem 6.4 quote: “Theorem 6.4. $K_2<0.3177$.”

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.