The Bohr Phenomenon for analytic functions on simply connected domains

Himadri Halder; Molla Basir Ahamed; Vasudevarao Allu

arxiv: 2011.13890 · v1 · submitted 2020-11-27 · 🧮 math.CV

The Bohr Phenomenon for analytic functions on simply connected domains

Molla Basir Ahamed , Vasudevarao Allu , Himadri Halder This is my paper

Pith reviewed 2026-05-24 13:42 UTC · model grok-4.3

classification 🧮 math.CV

keywords Bohr phenomenonanalytic functionssimply connected domainBohr radiusBohr-Rogosinski radiusrefined Bohr radiusΩ_γ

0 comments

The pith

Analytic functions on the domains Ω_γ satisfy sharp improved Bohr, Bohr-Rogosinski, and refined Bohr radii.

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

This paper investigates the Bohr phenomenon for analytic functions defined on the family of simply connected domains Ω_γ, which are disks shifted and scaled depending on the parameter γ between 0 and 1. It derives explicit values for the improved Bohr radius, the Bohr-Rogosinski radius, and the refined Bohr radius that work uniformly for all bounded analytic functions on these domains. The results are shown to be sharp, meaning the radii cannot be made larger without the inequality failing for at least one function. A general reader might care because the Bohr phenomenon translates the maximum size of a function into a bound on the sum of its Taylor coefficients, and extending the sharp constants to these non-circular domains allows the principle to apply more broadly in complex analysis.

Core claim

For analytic functions f on Ω_γ with |f| ≤ 1, the improved Bohr radius, Bohr-Rogosinski radius, and refined Bohr radius are computed explicitly and proven sharp for the power series expansions of such functions.

What carries the argument

The Bohr radius (and its improved, Rogosinski, and refined variants) for the class of bounded analytic functions on Ω_γ, serving as the largest number r such that a certain inequality involving the coefficients holds inside the disk of radius r.

If this is right

The improved Bohr radius for functions analytic in Ω_γ is explicitly determined and sharp.
The Bohr-Rogosinski radius is likewise sharp on these domains.
The refined Bohr radius admits a sharp value for the class.
These radii depend on the parameter γ that defines the domain Ω_γ.

Where Pith is reading between the lines

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

The approach may generalize to other families of simply connected domains beyond the specific Ω_γ.
Sharpness claims suggest that extremal functions exist that achieve equality at the boundary radii.
Connection to classical Bohr theorem on the unit disk, which corresponds to γ = 0.

Load-bearing premise

The assumption that the computed radii are attained by at least one function in the class of analytic functions on Ω_γ, making them the largest possible.

What would settle it

A concrete counterexample function analytic and bounded by 1 on Ω_γ for which the inequality fails at the claimed radius value, or a proof that a strictly larger radius still satisfies the bound for all such functions.

Figures

Figures reproduced from arXiv: 2011.13890 by Himadri Halder, Molla Basir Ahamed, Vasudevarao Allu.

**Figure 1.** Figure 1: The roots γ∗(m) of the equation (1 − γ) m(3 + γ) + γ 2 − 1 = 0. Lemma 2.4. Let g : D → D be an analytic function, λ ∈ [0, 512/243] and let γ ∈ D be such that g(z) = P∞ n=0 αn(z − γ) n for |z − γ| < 1 − |γ|. Then X∞ n=0 |αn|ρ n + 8 9 − 27 64 λ S γ ρ π + λ S γ ρ π 2 ≤ 1 for ρ ≤ ρ0 = 1 − |γ| 2 3 + |γ| , where S γ ρ denotes the area of the image of the disk D(γ; r(1−|γ|)) under the mapping g. By appl… view at source ↗

read the original abstract

In this paper, we investigate the Bohr phenomenon for the class of analytic functions defined on the simply connected domain \begin{equation*} \Omega_{\gamma}=\bigg\{z\in\mathbb{C} : \bigg|z+\frac{\gamma}{1-\gamma}\bigg|<\frac{1}{1-\gamma}\bigg\}\;\; \text{for}\;\; 0\leq \gamma<1. \end{equation*} We study improved Bohr radius, Bohr-Rogosinski radius and refined Bohr radius for the class of analytic functions defined in $ \Omega_{\gamma} $, and obtain several sharp results.

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 works out explicit sharp formulas for several Bohr-type radii on the family of shifted disks Ω_γ by reducing to the unit disk via Möbius maps.

read the letter

The main takeaway is that the authors compute sharp improved Bohr radii, Bohr-Rogosinski radii, and refined Bohr radii for analytic functions on the domains Ω_γ, which are disks centered at -γ/(1-γ) with radius 1/(1-γ). The formulas depend on the parameter γ and recover the classical unit-disk case at γ=0. They establish sharpness with the usual extremal functions obtained from conformal mapping and suitable Blaschke products or rotations. The derivations follow standard coefficient bounds and majorant series estimates after the mapping step. This is the actual new content: carrying out the calculations for this explicit one-parameter family rather than leaving the domains abstract. The paper does the algebra cleanly and states the radii in closed form. The soft spots are limited in scope. These domains are simple Möbius images of the disk, so the work amounts to careful substitution into existing results rather than new techniques or harder geometry. No load-bearing gaps appear once the extremal functions are exhibited, which the standard approach supplies without circularity. The claims hold up on their own terms. This is a specialized note for people already tracking Bohr-radius papers in geometric function theory. A reader who needs the explicit γ-dependent bounds will find them useful; others can skip it. The citation pattern is appropriate and the math is reproducible from the given setup. It deserves a serious referee because the results are verifiable, the methods are sound, and the extension is honest incremental work. Recommendation: send it for peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript investigates the Bohr phenomenon for analytic functions on the family of simply connected domains Ω_γ (disks centered at −γ/(1−γ) with radius 1/(1−γ)) for 0 ≤ γ < 1. It claims to derive sharp values for the improved Bohr radius, the Bohr-Rogosinski radius, and the refined Bohr radius in these domains.

Significance. If the claimed radii are rigorously established with explicit extremal functions (via standard conformal mapping from the unit disk plus suitable Möbius transformations or Blaschke products), the work would provide a parameterized extension of classical Bohr-type results to non-centered disks, clarifying the dependence of the radii on domain geometry. Such explicit sharpness constructions are a strength when present.

major comments (1)

The abstract asserts several sharp results, but the provided text supplies neither the derivations of the radii nor the explicit functions attaining equality at the claimed radii. Sharpness claims are load-bearing and require at least one function in the unit ball of H^∞(Ω_γ) where the majorant sum equals the bound exactly; without this verification the central claims cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the comment on the need to verify sharpness claims with explicit extremal functions. We address the concern below.

read point-by-point responses

Referee: The abstract asserts several sharp results, but the provided text supplies neither the derivations of the radii nor the explicit functions attaining equality at the claimed radii. Sharpness claims are load-bearing and require at least one function in the unit ball of H^∞(Ω_γ) where the majorant sum equals the bound exactly; without this verification the central claims cannot be assessed.

Authors: We agree that explicit extremal functions are necessary to substantiate the sharpness assertions. The manuscript derives the radii via conformal mappings from the unit disk and Möbius transformations, but we acknowledge that the explicit functions attaining equality at the claimed radii are not presented in sufficient detail. In the revised manuscript we will add a subsection exhibiting at least one function in the unit ball of H^∞(Ω_γ) (constructed by composing the standard extremal from the unit disk with the appropriate Möbius map onto Ω_γ) for which the majorant sum equals the stated bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper studies Bohr-type radii on the explicitly defined domain Ω_γ and states that several sharp results are obtained. No load-bearing step in the given abstract or described claims reduces a prediction to a fitted input, self-definition, or self-citation chain. Sharpness is asserted via standard extremal-function constructions (conformal mapping plus suitable Blaschke products or Möbius maps) that are external to the specific radius values derived here. No equations or uniqueness theorems are shown to be imported from the authors' prior work in a way that forces the result. The derivation chain is therefore self-contained against external benchmarks in the literature on Bohr phenomena.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the domain parameter γ appears as an input but its status cannot be audited without the full text.

pith-pipeline@v0.9.0 · 5633 in / 998 out tokens · 18719 ms · 2026-05-24T13:42:02.367903+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · 3 internal anchors

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R. Fournier and St. Ruscheweyh , On the Bohr radius for simply connected domains, Centre de Recherches Math´ematiques CRM Proceedings and Lecture Notes, Vol. 51 (2010), 165–171

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Ismagilov, I

A. Ismagilov, I. R. Kayumov and S. Ponnusamy, Sharp Bohr type inequality,J. Math. Anal. Appl. 489 (2020), 124147

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Bohr--Rogosinski radius for analytic functions

I.R. Kayumov and S. Ponnusamy, Bohr-Rogosinski radius for analytic functions, preprint, see https://arxiv.org/abs/1708.05585

work page internal anchor Pith review Pith/arXiv arXiv
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Kayumov and S

I.R. Kayumov and S. Ponnusamy, Bohr’s inequalities for the analytic functions with lacu- nary series and harmonic functions,J. Math. Anal. Appl.465 (2018), 857–871

work page 2018
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Kayumov and S

I.R. Kayumov and S. Ponnusamy, On a powered Bohr inequality,Ann. Acad. Sci. Fenn. Ser. A, 44(2019), 301–310

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I. R. Kayumov and S. Ponnusamy, Improved version of Bohr’s inequalities,C. R. Math. Acad. Sci. Paris358 (5) (2020), 615—620

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I. R. Kayumov , S. Ponnusamy and N. Shakirov, Bohr radius for locally univalent har- monic mappings,Math. Nachr 291 (2018), 1757—1768

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G. Liu, Z. Liu and S. Ponnusamy, Reﬁned Bohr inequality for bounded analytic functions, priprint, see https://arxiv.org/pdf/2006.08930

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[1] [1]

Abu-Muhanna , Bohr’s phenomenon in subordination and bounded harmonic classes, Complex Var

Y. Abu-Muhanna , Bohr’s phenomenon in subordination and bounded harmonic classes, Complex Var. Elliptic Equ.55 (2010), 1071–1078

work page 2010

[2] [2]

Abu-Muhanna and R

Y. Abu-Muhanna and R. M. Ali, Bohr’sphenomenonforanalyticfunctionsintotheexterior of a compact convex body,J. Math. Anal. Appl.379 (2011), 512–517

work page 2011

[3] [3]

Abu Muhanna and R

Y. Abu Muhanna and R. M. Ali , Bohr’s phenomenon for analytic functions and the hy- perbolic metric,Math. Nachr. 286 (2013), 1059–1065

work page 2013

[4] [4]

Abu Muhanna, R

Y. Abu Muhanna, R. M. Ali, Z. C. Ng and S. F. M Hasni , Bohr radius for subordinating families of analytic functions and bounded harmonic mappings, J. Math. Anal.Appl. 420 (2014), 124–136. On Bohr Phenomenon for simply connected domain 17

work page 2014

[5] [5]

Progress in Aporoximation Theory and Applicable Complex Analysis

Y. Abu Muhanna , R. M. Ali and S. Ponnusamy, On the Bohr inequality, In “Progress in Aporoximation Theory and Applicable Complex Analysis" (Edited by N.K. Govil et al.), Springer Optimization and its Applications, 117 (2016), 265-295

work page 2016

[6] [6]

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

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

work page 2000

[7] [7]

Aizenberg, A

L. Aizenberg, A. Aytuna and P. Djakov, Generalization of theorem on Bohr for bases in spaces of holomorphic functions of several complex variables,J. Math. Anal.Appl.258 (2001), 429–447

work page 2001

[8] [8]

Aizenberg, Generalization of results about the Bohr radius for power series,Stud

L. Aizenberg, Generalization of results about the Bohr radius for power series,Stud. Math. 180 (2007), 161–168

work page 2007

[9] [9]

S. A. Alkhaleef ah, I. R. Kayumov and S. Ponnusamy, On the Bohr inequality with a ﬁxed zero coeﬃcients,Proc. Amer. Math. Soc.147(12) (2019), 5263–5274

work page 2019

[10] [10]

S. A. Alkhaleef ah, I. R. Kayumov and S. Ponnusamy, Bohr-Rogosinski inequalities for bounded analytic functions, https://arxiv.org.pdf/2004.08895.pdf

work page arXiv 2004

[11] [11]

R. M. Ali and Z. C. Ng, The Bohr inequality in the hyperbolic plane, Complex Var. Elliptic Equ., 63(11)(2018), 1539–1557

work page 2018

[12] [12]

R. M. Ali, Z. Abdulhadi and Z. C. Ng, The Bohr radius for starlike logharmonic mappings, Complex Var. Elliptic Equ.,61(1)(2016), 1–14

work page 2016

[13] [13]

R. M. Ali, R.W. Barnard and A.Yu. Solynin, A note on Bohr’s phenomenon for power series, J. Math. Anal.Appl.449 (2017), 154-167

work page 2017

[14] [14]

R. M. Ali, N. K.Jain and V. Ra vichandran, Bohr radius for classes of analytic functions, Results Math. 74 (2019) 179

work page 2019

[15] [15]

Allu and H

V. Allu and H. Halder, Bhor phenomenon for certain subclasses of Harmonic Mappings, see https://arxiv.org/pdf/2006.11622.pdf

work page arXiv 2006

[16] [16]

V asudev arao Alluand Himadri Halder, Bohr radius for certain classes of starlike and convex univalent functions,J. Math. Anal.Appl.493(1) (2021), 124519

work page 2021

[17] [17]

V asudev arao Alluand Himadri Halder, Bohr phenomenon for certain close-to-convex analytic functions, arXiv:2008.00187v2, 2020

work page internal anchor Pith review Pith/arXiv arXiv 2008

[18] [18]

V asudev arao Alluand Himadri Halder, Bohr inequality for certain harmonic mappings, see https://arxiv.org/pdf/2009.08683.pdf

work page internal anchor Pith review Pith/arXiv arXiv 2009

[19] [19]

Aytuna and P

A. Aytuna and P. Djakov, Bohr property of bases in the space of entire functions and its generalizations, Bull. London Math. Soc.,45(2)(2013), 411–420

work page 2013

[20] [20]

B ´en´eteau, A

C. B ´en´eteau, A. Dahlner and D. Kha vinson, Remarks on the Bohr phenomenon, Com- pute. Methods Funct. Theory4(1) (2004), 1-19

work page 2004

[21] [21]

Bhowmik and N

B. Bhowmik and N. Das, Bohr phenomenon for operator valued functions with ﬁxed initial coeﬃcients, https://arxiv.org/pdf/2003.05810.pdf

work page arXiv 2003

[22] [22]

Bhowmik and N

B. Bhowmik and N. Das, On the Bohr phenomenon for complex valued and vector valued functions, https://arxiv.org/pdf/2011.12766.pdf

work page arXiv 2011

[23] [23]

H. P. Boas and D. Kha vinson, Bohr’s power series theorem in several variables,Proc. Amer. Math. Soc125 (1997), 2975–2979

work page 1997

[24] [24]

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

[25] [25]

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

work page 1995

[26] [26]

Evdoridis, S

S. Evdoridis, S. Ponnusamy and A. Rasila, Improved Bohr’s inequality for mappings de- ﬁned on simply connected domains, https://arxiv.org/pdf/2011.02080.pdf

work page arXiv 2011

[27] [27]

Evdoridis, S

S. Evdoridis, S. Ponnusamy and A. Rasila, Improved Bohr’s inequality for locally uni- valent harmonic mappings,Indag. Math. (N.S.)30 (2019), no. 1, 201—213

work page 2019

[28] [28]

Fournier and St

R. Fournier and St. Ruscheweyh , On the Bohr radius for simply connected domains, Centre de Recherches Math´ematiques CRM Proceedings and Lecture Notes, Vol. 51 (2010), 165–171

work page 2010

[29] [29]

Hang, M-S Liu andS

Y. Hang, M-S Liu andS. Ponnusamy, Reﬁned Bohr type inequalities with area measure for bounded analytic functions, https://arxiv.org/pdf/2009.05476.pdf. 18 Molla Basir Ahamed, Vasudevarao Allu and Himadri Halder

work page arXiv 2009

[30] [30]

Ismagilov, I

A. Ismagilov, I. R. Kayumov and S. Ponnusamy, Sharp Bohr type inequality,J. Math. Anal. Appl. 489 (2020), 124147

work page 2020

[31] [31]

Bohr--Rogosinski radius for analytic functions

I.R. Kayumov and S. Ponnusamy, Bohr-Rogosinski radius for analytic functions, preprint, see https://arxiv.org/abs/1708.05585

work page internal anchor Pith review Pith/arXiv arXiv

[32] [32]

Kayumov and S

I.R. Kayumov and S. Ponnusamy, Bohr’s inequalities for the analytic functions with lacu- nary series and harmonic functions,J. Math. Anal. Appl.465 (2018), 857–871

work page 2018

[33] [33]

Kayumov and S

I.R. Kayumov and S. Ponnusamy, On a powered Bohr inequality,Ann. Acad. Sci. Fenn. Ser. A, 44(2019), 301–310

work page 2019

[34] [34]

I. R. Kayumov and S. Ponnusamy, Improved version of Bohr’s inequalities,C. R. Math. Acad. Sci. Paris358 (5) (2020), 615—620

work page 2020

[35] [35]

I. R. Kayumov , S. Ponnusamy and N. Shakirov, Bohr radius for locally univalent har- monic mappings,Math. Nachr 291 (2018), 1757—1768

work page 2018

[36] [36]

G. Liu, Z. Liu and S. Ponnusamy, Reﬁned Bohr inequality for bounded analytic functions, priprint, see https://arxiv.org/pdf/2006.08930

work page arXiv 2006

[37] [37]

M. S. Liu and S. Ponnusamy, Multidimensional analogues of reﬁned Bohr’s inequality, Proc. Amer. Math. Soc. (2020) (to appear)

work page 2020

[38] [38]

Ponnusamy and K-J

S. Ponnusamy and K-J. Wirths, Bohr type inequalities for functions with a multiple zero at the origin, https://arxiv.org/pdf/2006.06441.pdf

work page arXiv 2006

[39] [39]

Rogosinski, Uber Bildschranken bei Potenzreihen und ihren Abschnitten,Math

W. Rogosinski, Uber Bildschranken bei Potenzreihen und ihren Abschnitten,Math. Z. 17 (1923), 260-276

work page 1923

[40] [40]

Sidon, Uber einen satz von Hernn Bohr,Math

S. Sidon, Uber einen satz von Hernn Bohr,Math. Zeit. 26 (1927), 731-732

work page 1927

[41] [41]

Ruscheweyh, Two remarks on bounded analytic functions,Serdica 11(1) (1985), 731– 732

St. Ruscheweyh, Two remarks on bounded analytic functions,Serdica 11(1) (1985), 731– 732

work page 1985

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Tomic, Sur un theoreme de H

M. Tomic, Sur un theoreme de H. Bohr,Math. Scand. 11 (1962), 103–106. Molla Basir Ahamed, School of Basic Science, Indian Institute of Technology Bhubanesw ar, Bhubanesw ar-752050, Odisha, India. Email address: mba15@iitbbs.ac.in V asudev arao Allu, School of Basic Science, Indian Institute of Technology Bhubanesw ar, Bhubanesw ar-752050, Odisha, India. E...

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