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arxiv: 2411.03352 · v1 · submitted 2024-11-04 · 🧮 math.CV

The Bohr's Phenomenon for the class of K-quasiconformal harmonic mappings

Raju Biswas , Rajib Mandal This is my paper

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classification 🧮 math.CV
keywords Bohr inequalityquasiconformal harmonic mappingssense-preserving mappingsunit diskBohr-Rogosinski inequalityrefined Bohr inequalitymajorant series
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The pith

K-quasiconformal sense-preserving harmonic mappings satisfy sharp improved Bohr inequalities in the unit disk.

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

The paper proves several sharp versions of the improved Bohr inequality, refined Bohr-type inequality, and refined Bohr-Rogosinski inequality for mappings of the form f = h + conj(g) that are K-quasiconformal and sense-preserving inside the unit disk. The proofs use a non-negative quantity S_ρ(h) and replace the initial coefficients in the majorant series by the absolute values of h and its derivative. A sympathetic reader would care because these results extend the classical Bohr radius phenomenon, which links coefficient sums to function growth, from analytic functions to a larger class of harmonic mappings with controlled distortion.

Core claim

For the class of K-quasiconformal sense-preserving harmonic mappings f = h + conj(g) in the unit disk, the authors establish sharp improved Bohr inequalities, refined Bohr-type inequalities, and refined Bohr-Rogosinski inequalities by employing the non-negative quantity S_ρ(h) and the replacement of initial coefficients by absolute values of the analytic function h and its derivative. They also obtain the sharp Bohr-Rogosinski radius for such harmonic mappings by replacing the usual bounding condition on h with the half-plane condition.

What carries the argument

The non-negative quantity S_ρ(h) combined with replacement of initial coefficients by |h| and |h'| in the majorant series.

If this is right

  • The Bohr-Rogosinski radius remains sharp when the bounding condition on the analytic part h is replaced by the half-plane condition.
  • Refined Bohr-type inequalities hold uniformly for the entire class with fixed quasiconformality constant K.
  • The same technique yields improved Bohr inequalities under the given replacement of coefficients by absolute values.

Where Pith is reading between the lines

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

  • The same replacement technique and S_ρ(h) construction may produce analogous radii for other bounded-distortion classes of harmonic mappings.
  • Numerical checks on explicit K-quasiconformal examples, such as affine stretches, could confirm whether the derived radii are attained.

Load-bearing premise

The mappings are sense-preserving and K-quasiconformal so that replacing initial coefficients by absolute values of h and h' remains valid while using the non-negative quantity S_ρ(h).

What would settle it

A concrete K-quasiconformal sense-preserving harmonic mapping f = h + conj(g) for which the majorant sum exceeds the claimed bound inside the stated radius would disprove the inequality.

Figures

Figures reproduced from arXiv: 2411.03352 by Rajib Mandal, Raju Biswas.

Figure 1
Figure 1. Figure 1: The graph of G8(ρ) In the following, we obtain the sharp version of Theorem F. Theorem 3.9. Suppose that f(z) = h(z) +g(z) = P∞ n=0 anz n + P∞ n=2 bnz n is a sense￾preserving K-quasiconformal harmonic mapping in D, where h(z) is bounded in D. Then X∞ n=0 |an|ρ n + X∞ n=2 |bn|ρ n ≤ ∥h(z)∥∞ for ρ ≤ ρ0, where ρ0 is the unique positive root of the equation 4ρ 1 − ρ  K K + 1 + 2  K − 1 K + 1 log(1 − ρ) = 1.… view at source ↗
read the original abstract

The primary objective of this paper is to establish several sharp versions of improved Bohr inequality, refined Bohr-type inequality, and refined Bohr-Rogosinski inequality for the class of $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| < 1\}$. In order to achieve these objectives, we employ the non-negative quantity $S_\rho(h)$ and the concept of replacing the initial coefficients of the majorant series by the absolute values of the analytic function and its derivative, as well as other various settings. Moreover, we obtain the sharp Bohr-Rogosinski radius for harmonic mappings in the unit disk by replacing the bounding condition on the analytic function $h$ with the half-plane condition.

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.

Referee Report

2 major / 0 minor

Summary. The paper claims to establish several sharp versions of the improved Bohr inequality, refined Bohr-type inequality, and refined Bohr-Rogosinski inequality for the class of K-quasiconformal sense-preserving harmonic mappings f = h + conj(g) in the unit disk. It employs the non-negative quantity S_ρ(h) together with the replacement of initial coefficients in the majorant series by absolute values of the analytic function h and its derivative, and additionally derives the sharp Bohr-Rogosinski radius under a half-plane condition on h.

Significance. If the central derivations hold, the work would extend the Bohr phenomenon from analytic functions to the broader class of K-quasiconformal harmonic mappings, supplying explicit sharp constants and radii that could serve as reference results in geometric function theory.

major comments (2)
  1. [Main results on improved and refined Bohr inequalities] The derivations of the improved Bohr, refined Bohr-type, and refined Bohr-Rogosinski inequalities rest on replacing the initial coefficients of the majorant series by |h| and |h'| via the non-negative quantity S_ρ(h). The manuscript does not supply an explicit verification that this replacement preserves validity and sharpness under the sole assumption that |g'/h'| ≤ k < 1; without such a step the claimed sharpness for the stated class cannot be confirmed.
  2. [Section on Bohr-Rogosinski radius] The argument for the sharp Bohr-Rogosinski radius obtained by replacing the bounding condition on h with the half-plane condition likewise invokes the same coefficient-replacement technique; the text provides no separate error estimate or counter-example check confirming that the K-quasiconformal dilatation bound continues to hold after the replacement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The observations highlight the need for greater explicitness in justifying the coefficient-replacement technique under the K-quasiconformal constraint. We address each point below and will incorporate the requested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Main results on improved and refined Bohr inequalities] The derivations of the improved Bohr, refined Bohr-type, and refined Bohr-Rogosinski inequalities rest on replacing the initial coefficients of the majorant series by |h| and |h'| via the non-negative quantity S_ρ(h). The manuscript does not supply an explicit verification that this replacement preserves validity and sharpness under the sole assumption that |g'/h'| ≤ k < 1; without such a step the claimed sharpness for the stated class cannot be confirmed.

    Authors: We agree that an explicit verification step would strengthen the exposition. The non-negativity of S_ρ(h) guarantees that the majorant series with coefficients replaced by |h| and |h'| remains a valid upper bound for the analytic part. Because the K-quasiconformal condition |g'/h'| ≤ k < 1 acts only on the conjugate part and does not alter the growth or coefficient bounds of h itself, the replacement preserves both the inequality and the sharpness attained by the standard extremal functions (e.g., suitable rotations of the Koebe-type mappings). In the revision we will insert a short lemma immediately before the main theorems that isolates this justification, citing the dilatation bound explicitly. revision: yes

  2. Referee: [Section on Bohr-Rogosinski radius] The argument for the sharp Bohr-Rogosinski radius obtained by replacing the bounding condition on h with the half-plane condition likewise invokes the same coefficient-replacement technique; the text provides no separate error estimate or counter-example check confirming that the K-quasiconformal dilatation bound continues to hold after the replacement.

    Authors: We accept the referee’s point that a separate verification is desirable. The half-plane condition is imposed solely on the analytic function h; the dilatation bound |g'/h'| ≤ k remains unchanged because g is determined by the same h via the quasiconformal relation. Consequently the radius obtained after replacement continues to be admissible for the full harmonic mapping. In the revised manuscript we will add a brief paragraph containing the error estimate (showing that the perturbation induced by the replacement is controlled by k) together with a remark confirming that the extremal function still satisfies the dilatation condition. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations rest on external complex-analysis facts

full rationale

The paper applies the non-negative quantity S_ρ(h) and coefficient-replacement technique to obtain Bohr-type inequalities for the stated class of K-quasiconformal harmonic mappings. These steps are presented as extensions of known methods rather than self-definitions or fitted inputs called predictions. No load-bearing self-citations, uniqueness theorems imported from the authors, or ansatzes smuggled via prior work are exhibited in the provided text. The central claims remain independent of the target results and do not reduce by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard background results from complex analysis and the theory of harmonic and quasiconformal mappings; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard properties of sense-preserving K-quasiconformal harmonic mappings in the unit disk
    Invoked to define the function class under study.

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Forward citations

Cited by 1 Pith paper

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

  1. A geometric investigation of a certain subclass of univalent functions

    math.CV 2024-11 unverdicted novelty 4.0

    Determines sharp radii for g h / z, z²/g(z), and z² / ∫(t/g(t)) dt to lie in M(λ) when g,h ∈ suitable subclasses of S, plus sharp Bohr, Bohr-Rogosinski, and improved Bohr radii for a subclass of starlike functions.

Reference graph

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