arxiv: 2604.14217 · v1 · submitted 2026-04-12 · 🧮 math.CV

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Bohr Radius and Landau-type Theorems for Harmonic Mappings with Boundary Functions in Lebesgue Spaces

Molla Basir Ahamed, Rajesh Hossain

Pith reviewed 2026-05-10 16:24 UTC · model grok-4.3

classification 🧮 math.CV

keywords Bohr radiusharmonic mappingsLebesgue spacesLandau theoremsPoisson integralunit diskunivalence radiusschlicht disk

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The pith

For harmonic mappings induced by L^p boundary functions on the circle, the Bohr radius is exactly 1/(2C_q + 1) and the univalence radius is bounded explicitly.

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

The paper proves a sharp Bohr-type inequality for bounded harmonic mappings in the unit disk whose analytic part satisfies a fixed coefficient condition. With |a0| equal to aM, the majorant series M_f(r) stays at most M inside the radius (1-a)/(1-a + 4/π), and this constant cannot be improved. The same approach yields an explicit sharp Bohr radius r_p = 1/(2C_q + 1) when the boundary function lies in L^p. Under standard normalization the work also supplies concrete formulas for the radius of univalence and the radius of the largest schlicht disk contained in the image. Sharpness in all cases is shown by mapping the Poisson kernel to suitable extremal examples.

Core claim

The central claim is that harmonic mappings f obtained via Poisson integral from boundary data F in L^p(T) obey the inequality M_f(r) ≤ M for r up to the value 1/(2C_q + 1), where C_q is determined by the conjugate exponent, and that this radius is best possible. The same framework produces explicit radii r0 of univalence and R0 of the inscribed schlicht disk, both attained by functions built from the Poisson kernel.

What carries the argument

The majorant series M_f(r) of the harmonic mapping together with its comparison to the bound M under the normalization |a0| = aM; this series controls growth and yields the Bohr radius via direct estimation against the Poisson integral representation.

If this is right

Inside the stated radius the growth of any such mapping is controlled by the boundary L^p norm alone.
The explicit univalence radius r0 guarantees local injectivity for every normalized mapping in the class.
The inscribed schlicht disk radius R0 gives a concrete lower bound on the size of the image.
Sharpness examples built from the Poisson kernel show that none of the radii can be enlarged without losing the inequality for all members of the class.

Where Pith is reading between the lines

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

The same Poisson-integral technique might produce Bohr radii for harmonic mappings whose boundary data lie in Orlicz or Lorentz spaces rather than L^p.
The derived radii could be compared numerically with the classical Bohr radius for analytic functions to quantify the enlargement or contraction caused by the harmonic extension.
Extremal Poisson-kernel examples suggest that similar sharp constants may hold for mappings defined on other domains whose boundary measures satisfy analogous integrability.

Load-bearing premise

The mappings must be exactly the Poisson integrals of their L^p boundary functions, together with the usual normalization conditions required for the univalence statements.

What would settle it

Exhibit one harmonic mapping whose boundary function is in L^p yet whose majorant series exceeds M at some radius strictly larger than 1/(2C_q + 1), or whose image fails to be univalent inside the claimed r0.

read the original abstract

This paper investigates the geometric and analytical properties of harmonic mappings $f$ in the unit disk $\mathbb{D}$ induced by boundary functions $F$ belonging to the Lebesgue spaces $L^{p}(\mathbb{T})$ for $1 \le p \le \infty$. We first establish a sharp Bohr-type inequality for the class of bounded harmonic mappings. Specifically, we prove that for a fixed analytic part $|a_{0}|= aM$, the majorant series $M_{f}(r)$ satisfies $M_{f}(r) \le M$ for $r \le (1-a)/(1-a+4/\pi)$, and demonstrate that this radius is best possible. This result is subsequently extended to harmonic mappings with $L^p$ boundary functions, where we determine the sharp Bohr radius $r_{p} = 1/(2C_{q}+1)$, with $C_{q}$ being a constant depending on the conjugate exponent $q$. Furthermore, the paper provides improved Landau-type theorems for these mappings. Under standard normalization, we derive explicit expressions for the radius of univalence $r_{0}$ and the radius of the inscribed schlicht disk $R_{0}$. The sharpness of these constants is discussed through the construction of extremal functions related to the Poisson kernel.

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

Gives explicit Bohr radii for L^p harmonic mappings but the sharpness claim for the bounded case rests on a majorant that the actual extremals do not attain.

read the letter

The paper derives an explicit Bohr radius (1-a)/(1-a + 4/π) for bounded harmonic mappings with fixed |a0| = aM and then extends it to L^p boundary data via r_p = 1/(2C_q + 1). It also supplies explicit radii of univalence and schlicht disks under standard normalization. These formulas are the concrete new pieces; the 4/π constant comes from the sharp coefficient bound |a_n| + |b_n| ≤ 4M/π obtained from the Poisson integral of ±M sgn(cos θ), and the L^p version follows from similar integral estimates on the conjugate exponent. The work is a direct extension of classical Bohr and Landau results to a larger class, and the authors construct extremals from the Poisson kernel to discuss sharpness. That is useful for specialists who want ready-to-use constants rather than existence statements. The main soft spot is the sharpness argument for the bounded case. The geometric-series majorant aM + (4M/π)r/(1-r) exceeds M only at a larger radius than claimed, and the Poisson-kernel extremals themselves have majorant (4M/π) artanh(r) that stays below M until roughly tanh(π/4) ≈ 0.655. No other function is exhibited that forces violation immediately past the stated radius, so the “best possible” assertion appears to rely on an unverified assumption that the coefficient bound can be approached in a way that saturates the majorant at the smaller value. The L^p formula looks more straightforwardly derived from the estimates and does not show the same gap. This is specialized work aimed at researchers already working on harmonic mappings and Bohr-type phenomena in geometric function theory. A reader who needs explicit radii for further estimates or comparisons will get value from the formulas, provided they check the sharpness proofs themselves. The paper is coherent on its own terms and engages the literature, so it deserves a serious referee rather than a desk rejection. The referee should focus on whether the bounded-case sharpness is actually established or whether the radius needs adjustment to match what the extremals deliver.

Referee Report

2 major / 2 minor

Summary. The paper claims to prove a sharp Bohr-type inequality for bounded harmonic mappings f in the unit disk with fixed analytic part |a0|=aM, showing that the majorant series M_f(r) ≤ M holds for r ≤ (1-a)/(1-a + 4/π) and that this radius is best possible via Poisson-kernel extremals; it extends the result to harmonic mappings induced by L^p(T) boundary functions (1≤p≤∞) with sharp radius r_p=1/(2C_q +1), and derives explicit radii r0 for univalence and R0 for the inscribed schlicht disk under standard normalization, again using Poisson-kernel constructions for sharpness.

Significance. If the sharpness assertions can be rigorously established, the results would provide explicit, parameter-dependent sharp constants for Bohr phenomena and Landau-type theorems in the setting of harmonic mappings with Lebesgue-integrable boundary data, extending classical analytic-function results to a broader class and offering concrete tools for geometric function theory.

major comments (2)

[Abstract / main Bohr theorem] Abstract and the statement of the main Bohr-radius theorem: the claim that r=(1-a)/(1-a+4/π) is best possible is not supported by the indicated extremals. The Poisson integral of ±M sgn(cos θ) yields a majorant (4M/π) artanh(r) that first exceeds M only at r=tanh(π/4)≈0.655, strictly larger than the asserted radius ≈0.440; the geometric-series majorant aM+(4M/π)r/(1-r) is valid but unattained by any single function, so an additional argument is required to show that the bound can be approached arbitrarily closely at the claimed radius.
[L^p extension section] Extension to L^p boundary functions: the constant C_q appearing in r_p=1/(2C_q+1) is not defined explicitly in the provided abstract, and the sharpness argument must be checked against the same Poisson-kernel construction; if C_q is obtained from an integral estimate independent of the target radius, the reduction to a geometric majorant again risks the same unattainability issue identified above.

minor comments (2)

Clarify the precise definition of the majorant series M_f(r) (sum of absolute coefficients or other form) and its relation to the analytic part a0.
Add explicit references to the classical Bohr theorem and to prior Landau-type results for harmonic mappings to situate the new constants.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and insightful comments on our manuscript. The points raised regarding the sharpness of the Bohr radii are important, and we address them point by point below. We will incorporate revisions to clarify and strengthen the proofs as needed.

read point-by-point responses

Referee: [Abstract / main Bohr theorem] Abstract and the statement of the main Bohr-radius theorem: the claim that r=(1-a)/(1-a+4/π) is best possible is not supported by the indicated extremals. The Poisson integral of ±M sgn(cos θ) yields a majorant (4M/π) artanh(r) that first exceeds M only at r=tanh(π/4)≈0.655, strictly larger than the asserted radius ≈0.440; the geometric-series majorant aM+(4M/π)r/(1-r) is valid but unattained by any single function, so an additional argument is required to show that the bound can be approached arbitrarily closely at the claimed radius.

Authors: We appreciate the referee's careful analysis of the extremal functions. Indeed, the specific Poisson integral of ±M sgn(cos θ) produces the majorant (4M/π) artanh(r), which remains below M up to a larger radius. Our claimed radius is obtained by bounding the majorant series using the geometric series sum aM + (4M/π) r/(1-r), leading to the smaller radius (1-a)/(1-a + 4/π). While this bound is valid for all such mappings, it is not attained by the indicated extremal. To rigorously establish that the radius is best possible, we will include in the revision an argument showing that the majorant can be made arbitrarily close to the geometric bound by considering suitable sequences of harmonic mappings whose Fourier coefficients approximate the worst-case scenario. This will demonstrate that for any r larger than the claimed value, there exists a function in the class where M_f(r) > M. We will revise the manuscript accordingly to provide this additional justification. revision: yes
Referee: [L^p extension section] Extension to L^p boundary functions: the constant C_q appearing in r_p=1/(2C_q+1) is not defined explicitly in the provided abstract, and the sharpness argument must be checked against the same Poisson-kernel construction; if C_q is obtained from an integral estimate independent of the target radius, the reduction to a geometric majorant again risks the same unattainability issue identified above.

Authors: We will ensure that the constant C_q is explicitly defined in the abstract of the revised manuscript (it is introduced in Section 3 as the constant arising from the relevant integral estimate involving the conjugate exponent q). For the sharpness argument, we will verify it against the Poisson-kernel construction and include an additional justification similar to the main theorem to demonstrate that the geometric majorant bound can be approached arbitrarily closely using appropriate sequences of L^p boundary functions. This will resolve the potential unattainability concern. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses independent coefficient bounds and extremal constructions

full rationale

The claimed Bohr radius follows from the standard coefficient estimate |a_n| + |b_n| ≤ 4M/π (obtained via Poisson integral of the sign function) followed by summation of the geometric majorant series, which is a direct algebraic consequence independent of the final radius value. Sharpness is asserted via explicit extremal functions constructed from the Poisson kernel, again without reducing the target radius to a fitted or self-referential quantity. No self-citations are load-bearing for the central inequality, no ansatz is smuggled, and no parameter is fitted to a subset then relabeled as a prediction. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper rests on standard representation theorems for harmonic functions and properties of Lebesgue spaces on the circle. No free parameters are fitted to data and no new entities are introduced.

axioms (2)

domain assumption Harmonic mappings in the unit disk admit a Poisson-integral representation from their boundary values in L^p(T).
Classical result in potential theory invoked to define the class of mappings studied.
standard math Coefficient majorants and univalence radii can be controlled via integral means and extremal Poisson-kernel examples.
Standard technique in geometric function theory for obtaining sharp constants.

pith-pipeline@v0.9.0 · 5536 in / 1426 out tokens · 55724 ms · 2026-05-10T16:24:12.283018+00:00 · methodology

discussion (0)

Reference graph

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