Bohr type inequality for certain integral operators and Fourier transform on shifted disks

Rajib Mandal; Raju Biswas; Vasudevarao Allu

arxiv: 2411.01674 · v1 · submitted 2024-11-03 · 🧮 math.CV

Bohr type inequality for certain integral operators and Fourier transform on shifted disks

Vasudevarao Allu , Raju Biswas , Rajib Mandal This is my paper

Pith reviewed 2026-05-23 17:41 UTC · model grok-4.3

classification 🧮 math.CV

keywords Bohr inequalityCesàro operatorBernardi integral operatordiscrete Fourier transformshifted disksbounded analytic functionsgeometric function theory

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The pith

Cesàro, Bernardi integral and discrete Fourier operators satisfy sharp Bohr-type inequalities for bounded analytic functions on shifted disks Ω_γ.

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

The paper derives sharp Bohr type inequalities for the Cesàro operator, the Bernardi integral operator, and the discrete Fourier transform applied to bounded analytic functions defined on shifted disks Ω_γ. These inequalities bound the sums of the absolute values of the Taylor coefficients of the resulting functions. A reader would care because the bounds are claimed to be best possible and extend the classical Bohr phenomenon from the unit disk to these translated circular domains for every γ in [0,1).

Core claim

In this paper, we derive the sharp Bohr type inequality for the Cesàro operator, Bernardi integral operator, and discrete Fourier transform acting on the class of bounded analytic functions defined on shifted disks Ω_γ for γ∈[0,1).

What carries the argument

The shifted disks Ω_γ = {z : |z + γ/(1-γ)| < 1/(1-γ)} for γ ∈ [0,1), the domain on which the bounded analytic functions live and to which the three operators are applied.

Load-bearing premise

The bounded analytic functions on Ω_γ form a class on which the Cesàro, Bernardi, and discrete Fourier operators produce functions whose coefficient sums satisfy the stated sharp Bohr-type bounds.

What would settle it

Exhibit one bounded analytic function on a specific Ω_γ whose image under the Cesàro operator (or either of the other two) has coefficient sum exceeding the claimed sharp bound at the relevant radius.

Figures

Figures reproduced from arXiv: 2411.01674 by Rajib Mandal, Raju Biswas, Vasudevarao Allu.

**Figure 1.** Figure 1: The graphs of Cγ when γ = 0, 0.2, 0.4, 0.5, 0.7 Besides the Bohr radius, there is another concept known as the Rogosinski radius [30, 42]. It is described as follows: Let SN (z) := PN−1 n=0 anz n be the partial sum of f ∈ B(D) defined by f(z) = P∞ n=0 anz n . Then, |SN (z)| < 1 for all N ≥ 1 in the disk |z| < 1/2. Here 1/2 is sharp, known as Rogosinski radius. Kayumov and Ponnusamy [20] have introduced the… view at source ↗

**Figure 2.** Figure 2: The graph of −2 (1/(1 − ρ) + log(1 − ρ)/ρ) for ρ ∈ [0, 1) Therefore, ∂ ∂aφ1(a, ρ) is a monotonically decreasing function of a ∈ [0, 1] and it follows that ∂ ∂aφ1(a, ρ) ≥ ∂ ∂aφ1(1, ρ) = −2ρ − 3(1 − ρ) log(1 − ρ) (1 − ρ)ρ ≥ 0 for ρ ≤ ρ0, where ρ0(≈ 0.533589) is the positive root of the equation 3(1 − ρ) log(1 − ρ) + 2ρ = 0, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: The graph of 3(1 − ρ) log(1 − ρ) + 2ρ for ρ ∈ [0, 1) Therefore φ1(a, ρ) is a monotonically increasing function of a ∈ [0, 1] and it follows that φ1(a, ρ) ≤ φ1(1, ρ) = − 1 ρ log(1 − ρ) = 1 ρ log 1 (1 − ρ) for ρ ≤ ρ0. To prove the sharpness of the result, we consider the function f1(z) in Ωγ such that f1 = ψ ◦ Φ1, where Φ1 : Ωγ → D defined by Φ1(z) = γ + (1 − γ)z and ψ : D → D defined by ψ(z) = (a − z)/(1 − … view at source ↗

read the original abstract

In this paper, we derive the sharp Bohr type inequality for the Ces\'aro operator, Bernardi integral operator, and discrete Fourier transform acting on the class of bounded analytic functions defined on shifted disks \beas \Omega_{\gamma}=\left\{z\in\mathbb{C}:\left|z+\frac{\gamma}{1-\gamma}\right|<\frac{1}{1-\gamma}\right\}\quad\text{for}\quad\gamma\in[0,1).\eeas

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

The paper claims sharp Bohr bounds for Cesàro, Bernardi, and discrete Fourier operators on bounded functions on shifted disks Ω_γ, but the sharpness likely fails to carry over because the admissible class is strictly smaller than the usual H^∞ on the disk.

read the letter

The main point is that the authors derive what they call sharp Bohr-type inequalities for the Cesàro operator, the Bernardi integral operator, and the discrete Fourier transform acting on analytic functions with |f| ≤ 1 on the shifted disks Ω_γ for γ in [0,1). The domains are larger than the unit disk, so the function class is smaller than the standard one on D. They present this as the contribution in the abstract. What is new is the application of the Bohr framework to these three operators on this one-parameter family of domains rather than the unit disk alone. The discrete Fourier transform part is also less common in the Bohr literature. The paper follows the usual pattern in this corner of geometric function theory by working with coefficient sums or growth estimates for the operator images. That part is standard and the abstract states the results clearly enough. The soft spot is the sharpness claim. The stress-test concern holds up on the information given. Because Ω_γ properly contains the open unit disk, any extremal function that exceeds modulus 1 at even one point outside D but inside Ω_γ is inadmissible. The classical extremals for the unit-disk versions of these inequalities are typically Möbius transformations or their compositions that attain the bound inside D; many of them will violate the bound on the extra region of Ω_γ. If the paper simply reuses those functions or the same constants without new admissible extremals, the claimed sharp constants are too small for the restricted class. The abstract gives no indication that this adjustment was made. This is narrow work inside geometric function theory. Readers already tracking Bohr radii, integral operators, and domain variations will see the most value; it is unlikely to matter much outside that group. The citation pattern and proof details are not visible here, but the topic is established enough that the paper deserves a serious referee to check the derivations and whether the sharpness argument actually works on the larger domain.

Referee Report

1 major / 2 minor

Summary. The manuscript derives sharp Bohr-type inequalities for the Cesàro operator, Bernardi integral operator, and discrete Fourier transform acting on the class of analytic functions f with |f| ≤ 1 on the shifted disks Ω_γ = {z : |z + γ/(1-γ)| < 1/(1-γ)} for γ ∈ [0,1).

Significance. If the sharpness arguments are valid for the restricted class of functions bounded on the full Ω_γ (a proper superset of the unit disk), the results would extend classical Bohr inequalities to a parameterized family of domains and provide explicit constants for these specific operators. The work supplies explicit derivations and sharpness claims for three operators.

major comments (1)

[proofs of the main theorems (Cesàro, Bernardi, and Fourier cases)] The central sharpness claims (for all three operators) rest on extremal functions that must remain in the class |f|≤1 on all of Ω_γ. Because Ω_γ properly contains the open unit disk and the class is therefore a strict subclass of H^∞(D), standard extremals (Möbius maps or their compositions) used for the unit-disk case may violate |f|≤1 at points of Ω_γ outside D. The manuscript must exhibit or verify admissible extremals that attain the stated constants inside Ω_γ; otherwise the constants are not sharp for the stated class. This issue is load-bearing for every “sharp” statement in the main theorems.

minor comments (2)

[abstract] The equation defining Ω_γ in the abstract uses display math that could be typeset more cleanly for readability.
[preliminaries] A brief remark on how the discrete Fourier transform is defined on the coefficient sequence of functions analytic in Ω_γ would clarify the operator for readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the critical issue with the sharpness arguments. We address the concern point by point below.

read point-by-point responses

Referee: [proofs of the main theorems (Cesàro, Bernardi, and Fourier cases)] The central sharpness claims (for all three operators) rest on extremal functions that must remain in the class |f|≤1 on all of Ω_γ. Because Ω_γ properly contains the open unit disk and the class is therefore a strict subclass of H^∞(D), standard extremals (Möbius maps or their compositions) used for the unit-disk case may violate |f|≤1 at points of Ω_γ outside D. The manuscript must exhibit or verify admissible extremals that attain the stated constants inside Ω_γ; otherwise the constants are not sharp for the stated class. This issue is load-bearing for every “sharp” statement in the main theorems.

Authors: We agree with the referee that the sharpness claims require extremal functions belonging to the class |f|≤1 on the entire domain Ω_γ, not merely on the unit disk. The manuscript currently invokes standard extremals from the classical unit-disk setting without explicitly verifying that these functions remain bounded by 1 throughout Ω_γ. This is a substantive gap. In the revised version we will either (i) construct or identify admissible extremals that attain the claimed constants while satisfying |f|≤1 on all of Ω_γ, or (ii) qualify the sharpness statements if no such functions exist for the given operators. The revision will cover the Cesàro, Bernardi, and discrete Fourier cases uniformly. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The abstract claims derivation of sharp Bohr-type inequalities for Cesàro, Bernardi, and discrete Fourier operators on bounded analytic functions over Ω_γ. No equations, coefficient relations, fitted parameters, or self-citations appear in the provided text that reduce any claimed prediction or bound to an input by construction. The domain and class definitions are independent of the target inequalities, and no load-bearing step is shown to collapse via self-definition, renaming, or imported uniqueness. The skeptic concern addresses whether sharpness holds for the restricted class, which is a correctness question outside circularity analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities.

pith-pipeline@v0.9.0 · 5601 in / 973 out tokens · 39050 ms · 2026-05-23T17:41:34.276798+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.1: Cf(ρ) ≤ (1/ρ)log(1/(1−ρ)) for |γ+(1−γ)z|=ρ≤ρ₀ where ρ₀ solves 3(1−ρ)log(1−ρ)+2ρ=0
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 2.2: |a_n| ≤ (1−γ)^n (1−|a_0|^2) from composition with Φ: D→Ω_γ

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · 1 internal anchor

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[1] [1]

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(4) (2000), 1147-1155

work page 2000

[2] [2]

S. A. Alkhaleefah, I. R. Kayumov and S. Ponnusamy , On the Bohr inequality with a fixed zero coefficient, Proc. Amer. Math. Soc. 147 (2019), 5263-5274

work page 2019

[3] [3]

Allu and H

V. Allu and H. Halder , Bohr radius for certain classes of starlike and convex univalent func- tions, J. Math. Anal. Appl. 493(1) (2021), 124519. 12 V. ALLU, R. BISWAS AND R. MANDAL

work page 2021

[4] [4]

Allu and H

V. Allu and H. Halder , Bohr phenomenon for certain subclasses of harmonic mappings, Bull. Sci. Math. 173 (2021), 103053

work page 2021

[5] [5]

Allu and H

V. Allu and H. Halder, Bohr phenomenon for certain close-to-convex analytic functions, Com- put. Methods Funct. Theory 22 (2022), 491-517

work page 2022

[6] [6]

Allu and H

V. Allu and H. Halder , Bohr inequality for certain harmonic mappings, Indag. Math. 33(3) (2022), 581-597

work page 2022

[7] [7]

Allu and H

V. Allu and H. Halder , Operator valued analogues of multidimensional Bohr’s inequality, Canadian Math. Bull. 65(4) (2022), 1020-1035

work page 2022

[8] [8]

Allu and V

V. Allu and V. Arora , Bohr-Rogosinski type inequalities for concave univalent functions, J. Math. Anal. Appl. 520 (2023), 126845

work page 2023

[9] [9]

Allu and N

V. Allu and N. Ghosh , Bohr type inequality for Ces´ aro and Bernardi integral operator on simply connected domain, Proc. Math. Sci. 133 (2023), 22

work page 2023

[10] [10]

V. Allu, H. Halder and S. Pal , Bohr and Rogosinski inequalities for operator valued holo- morphic functions, Bull. Sci. Math. 182 (2023), 103214

work page 2023

[11] [11]

Bachman and L

G. Bachman and L. Narici, Fourier and wavelet analysis, Springer-Verlag586, Springer, Berlin, 2000

work page 2000

[12] [12]

Bohr, A theory concerning power series, Proc

H. Bohr, A theory concerning power series, Proc. London Math. Soc. 13(2) (1914), 1-5

work page 1914

[13] [13]

Boas and D

H. Boas and D. Khavinson, Bohr’s power series theorem in several variables,Proc. Amer. Math. Soc. 125(10) (1997), 2975-2979

work page 1997

[14] [14]

Brawn, P

A. Brawn, P. R. Halmos and A. L. Shields , Ces´ aro operators,Acta Sci. Math. (Szeged) 26 (1965), 125-137

work page 1965

[15] [15]

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

work page 1995

[16] [16]

Evdoridis, S

S. Evdoridis, S. Ponnusamy and A. Rasila , Improved Bohr’s inequality for shifted disks, Results Math. 76 (2021), 14

work page 2021

[17] [17]

Fournier and ST

R. Fournier and ST. Ruscheweyh , On the Bohr radius for a simply connected plane domain, Centre de Recherches Mathematiques CRM Proceeding and Lecture Notes , 51 (2010), 165-171

work page 2010

[18] [18]

G. H. Hardy and J. E. Littlewood , Some properties of fractional integrals. II, Math. Z. 34 (1932), 403-439

work page 1932

[19] [19]

Ismagilov, I

A. Ismagilov, I. R. Kayumov and S. Ponnusamy, Sharp Bohr type inequality, J. Math. Anal. Appl. 489(1) (2020), 124147

work page 2020

[20] [20]

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

[21] [21]

I. R. Kayumov and S. Ponnusamy, Bohr inequality for odd analytic functions,Comput. Methods Funct. Theory 17 (2017), 679-688

work page 2017

[22] [22]

I. R. Kayumov and S. Ponnusamy , Improved version of Bohr’s inequality, C. R. Math. Acad. Sci. Paris 356(3) (2018), 272-277

work page 2018

[23] [23]

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

work page 2018

[24] [24]

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

work page 2019

[25] [25]

I. R. Kayumov, S. Ponnusamy and N. Shakirov , Bohr radius for locally univalent harmonic mappings, Math. Nachr. 11-12 (2018), 1757-1768

work page 2018

[26] [26]

I. R. Kayumov, D. M. Khammatova and S. Ponnusamy , On the Bohr inequality for the Ces´ aro operator,C. R. Math. Acad. Sci. Paris 358 (2020), 615-620

work page 2020

[27] [27]

I. R. Kayumov, D. M. Khammatova and S. Ponnusamy , Bohr-Rogosinski phenomenon for analytic functions and Ces´ aro operators,J. Math. Anal. Appl. 496 (2021), 124824

work page 2021

[28] [28]

I. R. Kayumov, D. M. Khammatova and S. Ponnusamy , The Bohr Inequality for the Gener- alized Ces´ aro Averaging Operators,Mediterr. J. Math. 19(19), (2022)

work page 2022

[29] [29]

Kumar and S

S. Kumar and S. K. Sahoo , Bohr inequalities for certain integral operators, Mediterr. J. Math. 18 (2021), 268

work page 2021

[30] [30]

Landau and D

E. Landau and D. Gaier, Darstellung und Begr¨ undung einiger neuerer Ergebnisse der Funktio- nentheorie (1986) (Berlin: Springer-Verlag)

work page 1986

[31] [31]

M. S. Liu and S. Ponnusamy , Multidimensional analogues of refined Bohr’s inequality, Proc. Amer. Math. Soc. 149 (2021), 2133-2146

work page 2021

[32] [32]

S. S. Miller and P. T. Mocanu , Differential Subordinations-Theory and Applications, Marcel Dekker, Inc., New York, 2000. BOHR TYPE INEQUALITY 13

work page 2000

[33] [33]

M. W. L. Ong and Z.C. Ng , On the Bohr Inequalities for Certain Integral Transforms, Iran J Sci (2024). https://doi.org/10.1007/s40995-024-01607-x

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