Multidimensional analogues of the improved Bohr's inequality for shifted polydisks

Abhijit Banerjee; Jhilik Banerjee; Shantanu Panja; Sujoy Majumder

Multidimensional analogues of the improved Bohr's inequality for shifted polydisks

T0 review · 2 major / 2 minor · reviewed 2026-07-15 · grok-4.5

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Pith's one-line read Improved Bohr radii for holomorphic functions on general simply connected domains in C^n and for pluriharmonic mappings on enlarged polydisks.

desk verdict Abstract-only multidimensional Bohr extension of Evdoridis et al.; honest subfield progress if the body holds, but nothing checkable yet. read the letter →

arxiv 2607.12777 v1 pith:VJF6WQCJ submitted 2026-07-14 math.CV

Sujoy Majumder , Abhijit Banerjee , Jhilik Banerjee , Shantanu Panja This is my paper · ORCID

classification math.CV MSC 32A0530C8032A10

keywords Bohrinequalitymultidimensionalradiusholomorphicfunctionspluriharmonicmappingspolydisksimplyconnecteddomainsseveralcomplexvariables

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends classical Bohr-type majorant inequalities from one complex variable into several variables. It claims that holomorphic functions on general simply connected domains in C^n obey improved Bohr radii of the same flavour that Evdoridis and co-authors obtained for shifted disks, but now for a strictly larger class of functions. Separately, it asserts that pluriharmonic mappings defined on a polydisk that properly contains the unit polydisk still satisfy a Bohr-type inequality. If the claims hold, they give concrete radius bounds that control the sum of the moduli of the coefficients by the supremum of the function, thereby enlarging the geometric settings in which such coefficient majorants are known to be valid.

What carries the argument

The multidimensional Bohr majorant inequality itself: an upper bound, valid inside a certain polydisk or domain of positive radius, that replaces the values of a holomorphic or pluriharmonic mapping by the sum of the moduli of its power-series coefficients, controlled by the supremum of the mapping.

What would settle it

An explicit holomorphic function on a simply connected domain in C^n (or a pluriharmonic mapping on a polydisk containing the unit polydisk) whose coefficient majorant exceeds the function's supremum inside every polydisk larger than the radius claimed by the paper.

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Extended reading notes

Core claim

Holomorphic functions belonging to a broader class than previously treated, defined on general simply connected domains in C^n, obey improved multidimensional Bohr inequalities that refine the earlier results of Evdoridis et al. for shifted disks; in addition, pluriharmonic mappings on any polydisk containing the unit polydisk satisfy a Bohr-type majorant inequality.

Load-bearing premise

The one-variable improved Bohr techniques of Evdoridis et al. extend to the claimed broader holomorphic class on simply connected domains in C^n and to pluriharmonic mappings on larger polydisks without extra hypotheses that would force the stated radii to shrink.

Editorial extensions

If this is right

Bohr radii previously known only for shifted disks become available for holomorphic functions on arbitrary simply connected domains in several complex variables.
The admissible class of holomorphic functions is strictly larger than the class treated by Evdoridis et al. (2021).
Pluriharmonic mappings on any polydisk containing the unit polydisk admit a Bohr-type majorant inequality.
Coefficient-sum estimates can be used on domains that properly contain the classical unit polydisk.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same majorant technique may adapt to other classes of mappings (e.g., harmonic or quasiregular) once corresponding one-variable Bohr radii are known.
If the radii turn out to be sharp, the paper would supply extremal examples that could serve as benchmarks for future multidimensional Bohr problems.
The results suggest a possible dictionary between geometric properties of the domain and the size of the Bohr radius in several complex variables.

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Signed reviews

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is an abstract-only look at a pure complex-analysis preprint on multidimensional Bohr inequalities. The one thing to know: it claims an improvement of Evdoridis et al. (Results Math. 76, 14, 2021) for a broader class of holomorphic functions on general simply connected domains in C^n, plus a Bohr-type inequality for pluriharmonic mappings on a polydisk containing the unit polydisk. That is the whole contribution as stated.

What is new, if true, is the multi-variable extension and the pluriharmonic piece. The program itself is established; this is a legitimate next step inside geometric function theory / several complex variables, not a new framework. Circularity looks low from the abstract—analytic majorant estimates building on prior work, no free parameters or invented entities visible. Pure-math inequality papers of this type are often sound when fully written, so I am not manufacturing a flaw.

The soft spot is purely informational. We cannot see the precise function-class definitions, the radii, the lemmas, or the comparison with the broader literature. The stress-test is right: the risk is that the one-variable improved Bohr techniques do not extend without extra restrictive hypotheses that shrink the stated radii. That is an information gap, not an identified internal inconsistency. Significance is limited-scope progress inside a specialized subfield; it does not reorganize anything.

Who it is for: people already working on Bohr radii, pluriharmonic mappings, and multi-variable geometric function theory. A general complex analyst will not get much from the abstract alone. I would not bring it to reading group until the body is available, and I would not cite it yet. It still deserves a serious referee if the full text appears with clean statements and proofs—desk-rejecting abstract-only extensions of this kind is premature. Send it to peer review when the manuscript is complete; expect the referee to check the multi-variable hypotheses carefully.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the Bohr phenomenon for holomorphic functions on general simply connected domains in C^n. It claims to improve the results of Evdoridis et al. (Results Math. 76, 14, 2021) for a broader class of holomorphic functions in several complex variables, and further establishes a Bohr-type inequality for pluriharmonic mappings defined on a polydisk containing the unit polydisk PΔ(0_n, 1_n).

Significance. If the claimed improvements hold with the stated generality and without hidden restrictions that shrink the radii, the work would extend the improved Bohr inequality of Evdoridis et al. from the one-variable shifted-disk setting to a multi-variable holomorphic class on simply connected domains in C^n and to pluriharmonic mappings on enlarged polydisks. Such extensions are of genuine interest in geometric function theory. The abstract alone does not allow confirmation of sharpness, explicit radii, or the breadth of the function class, so significance remains conditional on the body of the paper.

major comments (2)

[Manuscript body (unavailable)] Only the abstract is available for review. The central claims—an improvement of Evdoridis et al. (2021) for a broader holomorphic class on simply connected domains in C^n, and a Bohr-type inequality for pluriharmonic mappings on polydisks containing the unit polydisk—cannot be checked for correctness of proofs, precision of the radii obtained, or the presence of restrictive hypotheses that would limit the stated generality. A full assessment requires the complete manuscript with theorems, lemmas, and comparisons.
[Abstract (central claims)] The abstract asserts an extension of one-variable improved Bohr techniques to a broader multi-variable class and to pluriharmonic mappings without indicating whether additional hypotheses are imposed. Without the precise function-class definitions and the statements of the main theorems, it is impossible to verify that the multi-variable extension does not shrink the radii relative to the one-variable results being improved.

minor comments (2)

[Abstract] Typographical errors in the abstract: 'Bhor phenomenon' should read 'Bohr phenomenon'; 'Evdordis et al.' should read 'Evdoridis et al.' (consistent with the cited Results Math. 76, 14 (2021)).
[Abstract] Notation for the unit polydisk is written PΔ(0_n, 1_n); a brief clarification of this symbol (and consistency with standard polydisk notation) would aid readability once the full text is available.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detectable from abstract-only text; claimed improvements are presented as analytic extensions of prior Bohr inequalities.

full rationale

Only the abstract is available. It states that the authors investigate the Bohr phenomenon for holomorphic functions on general simply connected domains in C^n, improve results of Evdoridis et al. (2021) for a broader class, and establish a Bohr-type inequality for pluriharmonic mappings on a polydisk containing the unit polydisk. No equations, definitions of the function classes, fitted parameters, uniqueness theorems, or self-citations appear in the provided text. There is therefore no load-bearing step that can be shown to reduce by construction to its inputs, no fitted quantity renamed as a prediction, and no self-citation chain that forces the central claim. Standard citation of Evdoridis et al. is external prior work, not circular. Per the hard rules, an honest non-finding is required when the derivation cannot be inspected for reduction; score 0 with empty steps is the correct outcome. Any deeper circularity (or lack thereof) would require the full body.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

Abstract-only pure-math inequality paper. No numerical free parameters are indicated. Background axioms are standard complex analysis (holomorphy, simply connected domains, pluriharmonicity, polydisk geometry). No new physical or ad-hoc entities are introduced. The ledger is therefore thin; the real content would be the lemmas and majorant estimates in the missing body.

assumptions (3)

domain assumption Holomorphic functions on simply connected domains in C^n admit power-series expansions and majorant estimates of Bohr type.
Invoked by the abstract’s claim of Bohr phenomena for holomorphic functions on general simply connected domains in C^n.
domain assumption Pluriharmonic mappings on a polydisk containing the unit polydisk admit a Bohr-type majorant inequality.
Stated as the second main result; standard pluriharmonic theory is assumed as background.
domain assumption Results and techniques of Evdoridis et al., Results Math. 76, 14 (2021), on improved Bohr inequalities for shifted disks.
The abstract explicitly improves those results; they form the one-variable base case.

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Pith. "Pith review of Multidimensional analogues of the improved Bohr's inequality for shifted polydisks." pith.science (2026). https://pith.science/paper/VJF6WQCJ

@misc{pith2026260712777,
  author       = {Pith},
  title        = {Pith review of: Multidimensional analogues of the improved Bohr's inequality for shifted polydisks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VJF6WQCJ}},
  note         = {Machine review of arXiv:2607.12777}
}

abstract

In this article, we investigate the Bhor phenomenon for holomorphic functions defined on a general simply connected domain in $\mathbb{C}^n$. We improve the existing results Evdordis et al. (Improved Bohr's inequality for shifted disks, Results in Mathematics, 76, 14 (2021)) for a broader class of holomorphic functions in $\mathbb{C}^n$. Furthermore, we consider pluriharmonic mappings defined on a polydisk containing the unit polydisk $\mathbb{P}\Delta(0_n, 1_n)$ and establish a Bohr-type inequality for this class of mappings.

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Reviewed July 15, 2026 · model on record in the stance chip above.