Generalized Bohr inequalities for K-quasiconformal harmonic mappings and their applications

Rajib Mandal; Raju Biswas

arxiv: 2411.01837 · v1 · submitted 2024-11-04 · 🧮 math.CV

Generalized Bohr inequalities for K-quasiconformal harmonic mappings and their applications

Raju Biswas , Rajib Mandal This is my paper

Pith reviewed 2026-05-23 17:58 UTC · model grok-4.3

classification 🧮 math.CV

keywords Bohr inequalityquasiconformal mappingsharmonic mappingsunit diskmajorant seriesGaussian hypergeometric functionconvolution

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The pith

K-quasiconformal sense-preserving harmonic mappings on the unit disk satisfy several sharp generalized Bohr inequalities.

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

The paper establishes sharp versions of generalized Bohr inequalities for K-quasiconformal sense-preserving harmonic mappings on the unit disk. It replaces the usual power terms r^n with a majorant series built from any sequence of non-negative continuous functions ψ_n(r) whose sum converges locally uniformly on [0,1). The resulting bounds recover earlier Bohr results for harmonic mappings as special cases and also produce a convolution form of the theorem that involves the Gaussian hypergeometric function.

Core claim

For every K-quasiconformal sense-preserving harmonic mapping f on the unit disk and every admissible sequence {ψ_n(r)}, the majorant series satisfies a sharp inequality that bounds the modulus of f inside the disk; the same majorant construction yields refined and improved versions of classical Bohr statements together with a hypergeometric convolution identity that holds inside the same class.

What carries the argument

The majorant series formed by an arbitrary sequence {ψ_n(r)} of non-negative continuous functions whose sum converges locally uniformly on [0,1), applied to the class of K-quasiconformal sense-preserving harmonic mappings.

If this is right

All previously published Bohr inequalities for harmonic mappings become direct corollaries.
New sharply improved and refined Bohr inequalities hold for the full class of harmonic mappings on the disk.
A convolution counterpart of the Bohr theorem is obtained when the majorant series is realized by the Gaussian hypergeometric function.

Where Pith is reading between the lines

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

The same majorant technique could be tested on mappings that are merely harmonic but not quasiconformal to see whether the K-dependence is essential.
Special choices of ψ_n(r) might produce Bohr-type bounds for other function classes such as those with restricted coefficients.
The convolution identity suggests a possible link between Bohr radii and hypergeometric summation formulas that could be checked numerically for small K.

Load-bearing premise

The mappings must belong to the class of K-quasiconformal sense-preserving harmonic functions and the chosen sequence ψ_n(r) must consist of non-negative continuous functions whose series converges locally uniformly on [0,1).

What would settle it

An explicit K-quasiconformal sense-preserving harmonic mapping together with a concrete admissible sequence ψ_n(r) for which the claimed majorant inequality fails to hold with the stated constant or sharpness.

read the original abstract

The classical Bohr theorem and its subsequent generalizations have become active areas of research, with investigations conducted in numerous function spaces. Let $\{\psi_n(r)\}_{n=0}^\infty$ be a sequence of non-negative continuous functions defined on $[0,1)$ such that the series $\sum_{n=0}^\infty \psi_n(r)$ converges locally uniformly on the interval $[0, 1)$. The main objective of this paper is to establish several sharp versions of generalized Bohr inequalities for the class of $K$-quasiconformal sense-preserving harmonic mappings on the unit disk $\D := \{z \in \mathbb{C} : |z| < 1\}$. To achieve these, we employ the sequence of functions $\{\psi_n(r)\}_{n=0}^\infty$ in the majorant series rather than the conventional dependence on the basis sequence $\{r^n\}_{n=0}^\infty$. As applications, we derive a number of previously published results as well as a number of sharply improved and refined Bohr inequalities for harmonic mappings in $\D$. Moreover, we obtain a convolution counterpart of the Bohr theorem for harmonic mapping within the context of the Gaussian hypergeometric function

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 replaces the usual power majorant with a general convergent sequence to state Bohr inequalities for K-quasiconformal harmonic mappings and recovers earlier results as corollaries.

read the letter

The main point is a direct substitution: instead of the standard {r^n} they insert any sequence of non-negative continuous functions whose series converges locally uniformly on [0,1). This produces several claimed sharp inequalities for K-quasiconformal sense-preserving harmonic mappings on the disk, plus a convolution version that brings in the Gaussian hypergeometric function. The setup recovers a number of previously published bounds as special cases when the sequence is chosen appropriately. That unification is the clearest practical output. The conditions on the sequence are exactly the ones that usually make these majorant arguments go through, so the move is consistent with how similar generalizations have been handled elsewhere in the literature. The K-quasiconformal hypothesis is the natural one for the class they study. The soft spots are straightforward. The abstract states sharpness and applications but supplies none of the estimates or verification steps, so it is impossible to judge from the given material whether the bounds are attained or whether the quasiconformal dilatation introduces any extra looseness. The convolution result looks like a standard follow-on once the inequality is in place rather than an independent advance. This is a technical note aimed at the small group already working on Bohr phenomena for harmonic and quasiconformal mappings. A reader inside that subfield can extract the general statement and generate their own corollaries without much trouble. The argument is internally consistent on the terms described and shows ordinary engagement with the existing results. It is worth sending to a referee for a proper check of the derivations rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The paper claims to establish several sharp generalized Bohr inequalities for K-quasiconformal sense-preserving harmonic mappings on the unit disk by substituting an arbitrary sequence of non-negative continuous functions {ψ_n(r)} (with locally uniformly convergent series on [0,1)) in place of the standard majorant {r^n}. It derives applications recovering prior results, obtaining refined inequalities, and establishing a convolution counterpart of the Bohr theorem for such mappings in the setting of the Gaussian hypergeometric function.

Significance. If the derivations hold, the generalization unifies Bohr-type results across different majorants for this mapping class and recovers or sharpens existing bounds as special cases; the hypergeometric convolution application extends the framework to special functions, which may facilitate further work on coefficient problems and subordination in harmonic mappings.

minor comments (2)

[Abstract and §1] The abstract states that the inequalities are 'sharp' and that applications include 'sharply improved' bounds, but the manuscript should explicitly indicate in the introduction or §2 how sharpness is verified for the general {ψ_n(r)} (e.g., via extremal mappings or equality cases).
[§2] Notation for the class of K-quasiconformal harmonic mappings and the precise statement of the generalized majorant inequality should be introduced with a numbered display equation early in the paper to improve readability for readers unfamiliar with the specific sequence substitution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance of the generalization, and recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives generalized Bohr inequalities for K-quasiconformal sense-preserving harmonic mappings by substituting an arbitrary sequence {ψ_n(r)} (non-negative, continuous, locally uniformly convergent) into the majorant series in place of {r^n}. This substitution is justified directly by the stated hypotheses on the mapping class and on the sequence, which are the standard conditions under which such majorant replacements hold in the literature. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology; the central claims remain independent analytic estimates. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition and properties of K-quasiconformal sense-preserving harmonic mappings together with the convergence assumption on the majorant sequence; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption K-quasiconformal sense-preserving harmonic mappings satisfy the usual distortion and analyticity properties used in geometric function theory.
Invoked when defining the class to which the inequalities apply.
standard math The series sum ψ_n(r) converges locally uniformly on [0,1).
Stated explicitly as the condition on the majorant sequence.

pith-pipeline@v0.9.0 · 5736 in / 1252 out tokens · 19442 ms · 2026-05-23T17:58:01.283374+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.

Let {ψ_n(r)} be a sequence of non-negative continuous functions … such that ∑ ψ_n(r) converges locally uniformly on [0,1). … establish several sharp versions of generalized Bohr inequalities for the class of K-quasiconformal sense-preserving harmonic mappings
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 3.1 … Φ1(r) := 2K/(K+1) ∑ ψ_n(r) + … − p/2 ψ_0(r) ≤ 0 … |a0|^p ψ_0(r) + ∑ |an|ψ_n + ∑ |bn|ψ_n + G(∑ n|an|^2 ψ_n^2) ≤ ψ_0(r) ‖h‖_∞

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

48 extracted references · 48 canonical work pages · 1 internal anchor

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Abu-Muhanna, R

Y. Abu-Muhanna, R. M. Ali and S. Ponnusamy , On the Bohr inequality, In: N. K. Govil et al. (eds) Progress in approximation theory and applicable comp lex analysis, Springer Optimization and Its Applications, 117 (2017), 269–300

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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

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Aizenberg, A

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

work page 2001

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

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S. A. Alkhaleefah, I. R. Kayumov and S. Ponnusamy , Bohr-Rogosinski inequalities for bounded analytic functions, Lobachevskii J. Math. 41 (2020), 2110–2119

work page 2020

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M. B. Ahamed, V. Allu and H. Halder , The Bohr phenomenon for analytic functions on a shifted disk, Ann. Fenn. Math. 47 (2022), 103–120

work page 2022

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V. Allu, V. Arora and A. Shaji , On the second Hankel determinant of logarithmic coeﬃcient s for certain univalent functions, Mediterr. J. Math. 20 (2023), 81

work page 2023

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Allu and H

V. Allu and H. Halder , Bohr radius for certain classes of starlike and convex univ alent func- tions, J. Math. Anal. Appl. 493(1) (2021), 124519. GENERALIZED BOHR INEQUALITIES 25

work page 2021

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Allu and H

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

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Allu and H

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

work page 2022

[11] [11]

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

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Bisw as, Second Hankel determinant of logarithmic coeﬃcients for G(α) and P(M ), J

R. Bisw as, Second Hankel determinant of logarithmic coeﬃcients for G(α) and P(M ), J. Anal. (2024). https://doi.org/10.1007/s41478-024-00778-5

work page doi:10.1007/s41478-024-00778-5 2024

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B ´en´eteau, A

C. B ´en´eteau, A. Dahlner and D. Khavinson , Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4(1) (2004), 1–19

work page 2004

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

work page 1997

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Bohr , A theorem concerning power series, Proc

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

work page 1914

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Bombieri and J

E. Bombieri and J. Bourgain , A remark on Bohr’s inequality, Int. Math. Res. Not. 80 (2004), 4307–4330

work page 2004

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S. Y. Dai and Y. F. P an , Note on Schwarz-Pick estimates for bounded and positive re al part analytic functions, Proc. Am. Math. Soc. 136 (2008), 635–640

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Defant, L

A. Defant, L. Frerick, J. Ortega-Cerd `a, M. Ouna ¨ıes and K. Seip , The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontracti ve, Ann. Math. 174(2) (2011), 512–517

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Defant, D

A. Defant, D. Garc ´ıa, , M. Maestre and P. Sevilla-Peris , Dirichlet series and holomorphic functions in high dimension, 37, Cambridge University Press , 2019

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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

work page 2020

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Ismagilov, A

A. Ismagilov, A. V. Kayumova, I. R. Kayumov and S. Ponnusamy , Bohr inequalities in some classes of analytic functions, J. Math. Sci. 252(3) (2021), 360–373

work page 2021

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Kalaj , Quasiconformal harmonic mapping between Jordan domains, Math

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I. R. Kayumov, D. M. Khammatova and S. Ponnusamy , Bohr-Rogosinski phenomenon for analytic functions and Ces´ aro operators, J. Math. Anal. Appl. 493(2) (2021), 124824

work page 2021

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I. R. Kayumov, D. M. Khammatova and S. Ponnusamy , The Bohr inequality for the general- ized Ces´ aro averaging operators,Mediterr. J. Math. 19 (2022), 19

work page 2022

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I. R. Kayumov and S. Ponnusamy , Bohr-Rogosinski radius for analytic functions, preprint , https://doi.org/10.48550/arXiv.1708.05585

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1708.05585

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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

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

work page 2018

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

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S. G. Krantz , Geometric Function Theory. Explorations in Complex Analy sis. Birkh¨ auser, Boston, 2006

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