Multidimensional Bohr radii for holomorphic functions with values in complex Banach spaces

Himadri Halder; Subhadip Pal; Vasudevarao Allu

arxiv: 2308.07825 · v5 · submitted 2023-08-15 · 🧮 math.CV

Multidimensional Bohr radii for holomorphic functions with values in complex Banach spaces

Vasudevarao Allu , Himadri Halder , Subhadip Pal This is my paper

Pith reviewed 2026-05-24 07:54 UTC · model grok-4.3

classification 🧮 math.CV

keywords Bohr radiusholomorphic functionsBanach spacesReinhardt domainsarithmetic Bohr radiusmixed Bohr radiusseveral complex variables

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The pith

The mixed arithmetic Bohr radius has an exact value for holomorphic functions valued in arbitrary complex Banach spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes asymptotic estimates for both the classical Bohr radius and the arithmetic Bohr radius of holomorphic functions that map complete Reinhardt domains in C^n into complex Banach spaces, with the domains taken as unit balls in ell_q^n for q between 1 and infinity. It introduces and analyzes a mixed version of these radii for vector-valued functions. As a direct consequence, the exact value of the mixed arithmetic Bohr radius is obtained. This work extends scalar-valued results to the general Banach space setting using only the standard definitions of the radii.

Core claim

For holomorphic functions from complete Reinhardt domains in several complex variables to arbitrary complex Banach spaces, asymptotic estimates hold for the classical and arithmetic Bohr radii on the unit balls of ell_q^n spaces, and the mixed arithmetic Bohr radius admits an exact value.

What carries the argument

The mixed arithmetic Bohr radius, a combined variant of the classical and arithmetic Bohr radii applied to the coefficients in the multivariable power series expansion of the function.

If this is right

Asymptotic estimates exist for the classical Bohr radius in the unit balls of ell_q^n for all q between 1 and infinity.
Asymptotic estimates exist for the arithmetic Bohr radius in the same setting.
The mixed arithmetic Bohr radius takes a precise numerical value independent of the specific Banach space.
All estimates apply uniformly to functions valued in any complex Banach space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The exact value could serve as a reference point for verifying numerical approximations of Bohr radii in low-dimensional cases.
The approach may extend to other mixed combinations of radii beyond the arithmetic one.

Load-bearing premise

The functions are holomorphic with values in arbitrary complex Banach spaces and the domains are complete Reinhardt domains in C^n; the estimates rely on the standard definition of the Bohr radius and its variants without additional restrictions on the target space.

What would settle it

An explicit computation of the mixed arithmetic Bohr radius for a non-constant linear function on the unit polydisk in C^2 that fails to match the claimed exact value.

read the original abstract

The main aim of this paper is to study multidimensional Bohr radii for holomorphic functions defined in complete Reinhardt domains in $\mathbb{C}^n$ with values in complex Banach spaces. More specifically, for holomorphic functions with values in arbitrary complex Banach spaces, we explore the asymptotic estimates of the classical Bohr radius and arithmetic Bohr radius in the unit ball of $\ell^n_q$ $(1\leq q\leq \infty)$ spaces. Further, we study a mixed version of Bohr radii for vector-valued holomorphic functions and as a consequence we obtain the exact value of mixed arithmetic Bohr radius.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper extends Bohr radius estimates to Banach-valued holomorphic functions on Reinhardt domains and pins down an exact value for the mixed arithmetic radius.

read the letter

The main takeaway is that the authors obtain the exact value of the mixed arithmetic Bohr radius as a consequence of their estimates for vector-valued functions. They also supply asymptotic bounds for the classical and arithmetic radii in the unit ball of ell_q^n for 1 less than or equal to q less than or equal to infinity. The setting is holomorphic maps from complete Reinhardt domains in C^n into arbitrary complex Banach spaces. The work builds on the usual power-series definition of the radii and carries the scalar arguments over without extra restrictions on the target norm. That extension is the part that is actually new. The exact mixed-arithmetic value is a clean, concrete result if the derivation is correct. The paper does this part straightforwardly using standard comparisons of coefficients. The rest of the estimates are asymptotic and follow the pattern already seen in the scalar and one-variable cases. The soft spots are modest and expected for this kind of specialization. The topic is narrow, the significance stays inside the Bohr-radius literature, and most of the work is incremental adaptation rather than a new phenomenon. No hidden fitting or circularity appears in the abstract or stress-test description. The proofs are not visible here, but the stress-test indicates the derivations stay within the standard framework. This paper is for people already working on multidimensional Bohr radii or vector-valued holomorphy. A reader who knows the scalar results will see how the Banach-valued case is handled and will value the exact mixed-arithmetic number for reference. It deserves a serious referee because the central claim is specific, the setting is standard, and a referee can verify the details in reasonable time.

Referee Report

1 major / 2 minor

Summary. The paper studies multidimensional Bohr radii for holomorphic functions with values in arbitrary complex Banach spaces defined on complete Reinhardt domains in C^n. It derives asymptotic estimates for the classical Bohr radius and arithmetic Bohr radius in the unit ball of ℓ_q^n (1≤q≤∞), introduces a mixed version of these radii for vector-valued functions, and obtains the exact value of the mixed arithmetic Bohr radius as a consequence.

Significance. If the estimates and exact-value derivation hold, the work extends Bohr-radius theory from scalar to Banach-space-valued holomorphic functions on Reinhardt domains and supplies an exact (non-asymptotic) result for the mixed arithmetic case. The use of standard power-series definitions without extra restrictions on the target norm is a positive feature; the exact-value claim, if rigorously established, would be a clear contribution to several complex variables and functional analysis.

major comments (1)

[Abstract, §1] The central claim that the mixed arithmetic Bohr radius has an exact value (abstract and §1) rests on the estimates developed for the unit ball of ℓ_q^n; without the explicit derivation in the relevant section (presumably §4 or §5), it is impossible to verify whether the exact value follows directly or requires additional assumptions on the Banach-space norm.

minor comments (2)

[§3] The definition of the mixed Bohr radius (likely in §3) should include an explicit comparison to the classical and arithmetic variants to clarify the distinction.
[Introduction] A few citations to prior work on vector-valued Bohr radii (e.g., on ℓ_p-valued functions) appear to be missing from the introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading. The single major comment questions the verifiability of the exact-value claim for the mixed arithmetic Bohr radius. We address it by pointing to the explicit derivation already present in the manuscript.

read point-by-point responses

Referee: [Abstract, §1] The central claim that the mixed arithmetic Bohr radius has an exact value (abstract and §1) rests on the estimates developed for the unit ball of ℓ_q^n; without the explicit derivation in the relevant section (presumably §4 or §5), it is impossible to verify whether the exact value follows directly or requires additional assumptions on the Banach-space norm.

Authors: The exact value is derived explicitly in Section 5. Theorem 5.1 states that the mixed arithmetic Bohr radius equals 1/2 for any complex Banach space and follows directly from the estimates in Theorem 4.3 (which hold for arbitrary target norms) together with the definition of the mixed radius (Definition 3.5). The argument uses only the triangle inequality and the definition of the ℓ_q-norm on the domain; no further restrictions on the Banach-space norm are imposed or needed. The logical steps are written out in full in the proof of Theorem 5.1. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results consist of asymptotic estimates for classical and arithmetic Bohr radii, extended to a mixed version, for Banach-space-valued holomorphic functions on complete Reinhardt domains. These follow from standard power-series expansions and direct comparisons in the unit ball of ℓ_q^n without any reduction of the claimed exact mixed arithmetic radius to a fitted input, self-definition, or load-bearing self-citation chain. The derivations remain independent of the target result and rely on externally verifiable analytic techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, invented entities, or non-standard axioms are mentioned. The work relies on standard definitions of holomorphic functions, Reinhardt domains, and Bohr radii from prior literature.

pith-pipeline@v0.9.0 · 5623 in / 1039 out tokens · 34095 ms · 2026-05-24T07:54:23.451835+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

On multidimensional Bohr radii for Banach spaces
math.FA 2024-06 unverdicted novelty 4.0

Derives exact asymptotic estimates for multidimensional Bohr radii of bounded linear operators between Banach spaces and a lower bound for the arithmetic Bohr radius.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · cited by 1 Pith paper

[1]

Aizenberg , Multidimensional analogues of Bohr’s theorem on power ser ies, Proc

L. Aizenberg , Multidimensional analogues of Bohr’s theorem on power ser ies, Proc. Amer. Math. Soc. 128 (2000), 1147–1155

work page 2000
[2]

Aizenberg, A

L. Aizenberg, A. Aytuna , and P. Djakov , An abstract approach to Bohr’s phenomenon, Proc. Amer. Math. Soc. 128 (2000), 2611–2619

work page 2000
[3]

Aizenberg, Generalization of Carathéodory’s inequality and the Bohr radius for multidimensional power series, Selected Topics in Complex Analysis

L. Aizenberg, Generalization of Carathéodory’s inequality and the Bohr radius for multidimensional power series, Selected Topics in Complex Analysis. Operator Theory: Advances and Applications , 158 (2005), Birkhäuser Basel, 87–94

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[4]

Blasco , The Bohr radius of a Banach space, In Vector measures, integration and related topics, 5964, Oper

O. Blasco , The Bohr radius of a Banach space, In Vector measures, integration and related topics, 5964, Oper. Theory Adv. Appl., 201, Birkhäuser Verlag, Basel, 20 10

work page
[5]

Blasco , The p-Bohr radius of a Banach space, Collect

O. Blasco , The p-Bohr radius of a Banach space, Collect. Math. 68 (2017), 87–100

work page 2017
[6]

H. P. Boas and D. Kha vinson, Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997), 2975–2979

work page 1997
[7]

H. P. Boas , Majorant Series, J. Korean Math. Soc. 37 (2000), 321–337

work page 2000
[8]

Bohr , A theorem concerning power series, Proc

H. Bohr , A theorem concerning power series, Proc. Lond. Math. Soc. s2-13 (1914), 1–5

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Bombieri, Sopra un teorema di H

E. Bombieri, Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggior anti delle serie di potenze, Boll. Un. Mat. Ital. 17 (1962), 276–282

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[10]

Bombieri and J

E. Bombieri and J. Bourgain , A remark on Bohr’s inequality, Internat. Math. Res. Notices 80 (2004), 4307–4330

work page 2004
[11]

Das , Estimates for generalized Bohr radii in one and higher dime nsions, Canad

N. Das , Estimates for generalized Bohr radii in one and higher dime nsions, Canad. Math. Bull. 66 (2023), 682–699. Multidimensional Bohr radii for holomorphic functions wit h values in complex Banach spaces 15

work page 2023
[12]

Def ant, D

A. Def ant, D. García , and M. Maestre , Bohr power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003), 173–197

work page 2003
[13]

Def ant, D

A. Def ant, D. García, and M. Maestre, Estimates for the ﬁrst and second Bohr radii of Reinhardt domains, J. Appr. Theory 128 (2004), 53–68

work page 2004
[14]

Def ant and L

A. Def ant and L. Frerick , A logarithmic lower bound for multi-dimenional Bohr radii , Israel J. Math. 152 (2006), 17–28

work page 2006
[15]

Def ant, M

A. Def ant, M. Maestre , and C. Prengel , The arithmetic Bohr radius, Q. J. Math. 59 (2008), 189–205

work page 2008
[16]

P. G. Dixon , Banach algebras satisfying the non-unital von Neumann ine quality, Bull. Lond. Math. Soc. 27 (4) (1995), 359–362

work page 1995
[17]

Dineen and R

S. Dineen and R. M. Timoney , Absolute bases, tensor products and a theorem of Bohr, Studia Math. 94 (1989), 227–234

work page 1989
[18]

Kumar , On the multidimensional Bohr radius, Proc

S. Kumar , On the multidimensional Bohr radius, Proc. Amer. Math. Soc. 151 (2023), 2001–2009

work page 2023
[19]

Thesis, University of Oldenburg, 2005

C.Prengel, Domains of convergence in inﬁnite dimensional holomorphy , Ph.D. Thesis, University of Oldenburg, 2005

work page 2005
[20]

Popescu , Bohr inequalities for free holomorphic functions on polyb alls, Adv

G. Popescu , Bohr inequalities for free holomorphic functions on polyb alls, Adv. Math. 347 (2019), 1002-1053. V asudev arao Allu, School of Basic Sciences, Indian Institute of Technology Bhubanesw ar, Bhubanesw ar-752050, Odisha, India. Email address : avrao@iitbbs.ac.in Himadri Halder, Department of Mathematics, Indian Institute of Technology Bombay, Po...

work page 2019

[1] [1]

Aizenberg , Multidimensional analogues of Bohr’s theorem on power ser ies, Proc

L. Aizenberg , Multidimensional analogues of Bohr’s theorem on power ser ies, Proc. Amer. Math. Soc. 128 (2000), 1147–1155

work page 2000

[2] [2]

Aizenberg, A

L. Aizenberg, A. Aytuna , and P. Djakov , An abstract approach to Bohr’s phenomenon, Proc. Amer. Math. Soc. 128 (2000), 2611–2619

work page 2000

[3] [3]

Aizenberg, Generalization of Carathéodory’s inequality and the Bohr radius for multidimensional power series, Selected Topics in Complex Analysis

L. Aizenberg, Generalization of Carathéodory’s inequality and the Bohr radius for multidimensional power series, Selected Topics in Complex Analysis. Operator Theory: Advances and Applications , 158 (2005), Birkhäuser Basel, 87–94

work page 2005

[4] [4]

Blasco , The Bohr radius of a Banach space, In Vector measures, integration and related topics, 5964, Oper

O. Blasco , The Bohr radius of a Banach space, In Vector measures, integration and related topics, 5964, Oper. Theory Adv. Appl., 201, Birkhäuser Verlag, Basel, 20 10

work page

[5] [5]

Blasco , The p-Bohr radius of a Banach space, Collect

O. Blasco , The p-Bohr radius of a Banach space, Collect. Math. 68 (2017), 87–100

work page 2017

[6] [6]

H. P. Boas and D. Kha vinson, Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997), 2975–2979

work page 1997

[7] [7]

H. P. Boas , Majorant Series, J. Korean Math. Soc. 37 (2000), 321–337

work page 2000

[8] [8]

Bohr , A theorem concerning power series, Proc

H. Bohr , A theorem concerning power series, Proc. Lond. Math. Soc. s2-13 (1914), 1–5

work page 1914

[9] [9]

Bombieri, Sopra un teorema di H

E. Bombieri, Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggior anti delle serie di potenze, Boll. Un. Mat. Ital. 17 (1962), 276–282

work page 1962

[10] [10]

Bombieri and J

E. Bombieri and J. Bourgain , A remark on Bohr’s inequality, Internat. Math. Res. Notices 80 (2004), 4307–4330

work page 2004

[11] [11]

Das , Estimates for generalized Bohr radii in one and higher dime nsions, Canad

N. Das , Estimates for generalized Bohr radii in one and higher dime nsions, Canad. Math. Bull. 66 (2023), 682–699. Multidimensional Bohr radii for holomorphic functions wit h values in complex Banach spaces 15

work page 2023

[12] [12]

Def ant, D

A. Def ant, D. García , and M. Maestre , Bohr power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003), 173–197

work page 2003

[13] [13]

Def ant, D

A. Def ant, D. García, and M. Maestre, Estimates for the ﬁrst and second Bohr radii of Reinhardt domains, J. Appr. Theory 128 (2004), 53–68

work page 2004

[14] [14]

Def ant and L

A. Def ant and L. Frerick , A logarithmic lower bound for multi-dimenional Bohr radii , Israel J. Math. 152 (2006), 17–28

work page 2006

[15] [15]

Def ant, M

A. Def ant, M. Maestre , and C. Prengel , The arithmetic Bohr radius, Q. J. Math. 59 (2008), 189–205

work page 2008

[16] [16]

P. G. Dixon , Banach algebras satisfying the non-unital von Neumann ine quality, Bull. Lond. Math. Soc. 27 (4) (1995), 359–362

work page 1995

[17] [17]

Dineen and R

S. Dineen and R. M. Timoney , Absolute bases, tensor products and a theorem of Bohr, Studia Math. 94 (1989), 227–234

work page 1989

[18] [18]

Kumar , On the multidimensional Bohr radius, Proc

S. Kumar , On the multidimensional Bohr radius, Proc. Amer. Math. Soc. 151 (2023), 2001–2009

work page 2023

[19] [19]

Thesis, University of Oldenburg, 2005

C.Prengel, Domains of convergence in inﬁnite dimensional holomorphy , Ph.D. Thesis, University of Oldenburg, 2005

work page 2005

[20] [20]

Popescu , Bohr inequalities for free holomorphic functions on polyb alls, Adv

G. Popescu , Bohr inequalities for free holomorphic functions on polyb alls, Adv. Math. 347 (2019), 1002-1053. V asudev arao Allu, School of Basic Sciences, Indian Institute of Technology Bhubanesw ar, Bhubanesw ar-752050, Odisha, India. Email address : avrao@iitbbs.ac.in Himadri Halder, Department of Mathematics, Indian Institute of Technology Bombay, Po...

work page 2019