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arxiv: 2606.02612 · v1 · pith:QHQPZIQXnew · submitted 2026-05-25 · 🧮 math.CV

Bohr, Bohr-Rogosinski, and Landau-Type Results for a Generalized Class of Harmonic Mappings

Xiaoyuan Wang , Xintong Han , Rajesh Hossain , Molla Basir Ahamed This is my paper

Pith reviewed 2026-06-29 19:50 UTC · model grok-4.3

classification 🧮 math.CV
keywords harmonic mappingsBohr phenomenonBohr radiusBohr-Rogosinski radiusLandau theoremdifferential inequalityunivalence radiuscoefficient estimates
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The pith

A generalized class of harmonic mappings defined by differential inequality satisfies refined Bohr radii and Landau theorems.

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

The paper defines the class BH_0(γ, δ) of harmonic mappings in the unit disk by a second-order differential inequality that extends known subclasses of harmonic and analytic functions. Using sharp coefficient estimates and growth results, the authors derive improved Bohr-type inequalities with refined radii, Bohr-Rogosinski radii, and inequalities for higher-order coefficient sums and area terms. They also obtain Landau-type theorems that give explicit bounds on the radius of univalence and the size of schlicht disks contained in the image. These results are shown to be sharp via extremal functions and unify several earlier findings for the harmonic setting.

Core claim

For functions in the class BH_0(γ, δ), sharp coefficient estimates yield explicit refined Bohr radii and Bohr-Rogosinski radii together with generalized inequalities involving higher-order coefficient sums and area terms; the same estimates also produce Landau-type theorems that bound the radius of univalence and the size of schlicht disks in the image domain, with all bounds verified as sharp by extremal functions in the class.

What carries the argument

The class BH_0(γ, δ) of harmonic mappings defined via a second-order differential inequality, combined with sharp coefficient estimates and growth results to obtain the radii and inequalities.

If this is right

  • The Bohr radius for BH_0(γ, δ) is refined and depends explicitly on the parameters γ and δ.
  • Bohr-Rogosinski radii hold for the same class.
  • Generalized inequalities are valid for sums of higher-order coefficients and area terms.
  • Explicit upper bounds are obtained for the radius of univalence of functions in the class.
  • The size of schlicht disks contained in the image is bounded in terms of the parameters.

Where Pith is reading between the lines

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

  • The same differential-inequality approach could be applied to obtain radii for other constrained classes of harmonic mappings.
  • The bounds might be checked numerically for concrete values of γ and δ to test sharpness in practice.
  • When the anti-holomorphic part vanishes, the results should recover corresponding statements for analytic functions satisfying the inequality.
  • The univalence-radius bounds could be compared with known distortion theorems for harmonic mappings to see whether tighter estimates emerge.

Load-bearing premise

The extremal functions invoked for sharpness actually belong to the class BH_0(γ, δ) and attain the stated coefficient and growth bounds.

What would settle it

Exhibiting a function in BH_0(γ, δ) for specific γ and δ whose coefficient sum or image schlicht disk size exceeds the claimed bound at the stated radius.

read the original abstract

In this paper, we study the Bohr phenomenon for a generalized subclass of harmonic mappings defined by a second-order differential inequality in the unit disk. Specifically, we consider the class $\mathcal{BH}_0(\gamma, \delta)$, which extends several known subclasses of harmonic and analytic functions. By employing sharp coefficient estimates and growth results, we establish improved versions of Bohr-type inequalities, including refined Bohr radii and Bohr--Rogosinski radii for this class. Furthermore, we derive generalized inequalities involving higher-order coefficient sums and area terms, thereby extending classical Bohr inequalities in a harmonic setting. The sharpness of the obtained results is verified through extremal functions. In addition, we obtain Landau-type theorems for the class $\mathcal{BH}_0(\gamma, \delta)$, providing explicit bounds for the radius of univalence and the size of schlicht disks contained in the image domain. Our results not only unify and extend several earlier works but also provide new insights into the geometric behavior of harmonic mappings under differential constraints.

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 introduces the class BH_0(γ, δ) of harmonic mappings in the unit disk defined via a second-order differential inequality. It claims to derive improved Bohr-type inequalities with refined Bohr and Bohr-Rogosinski radii, generalized bounds on higher-order coefficient sums and area terms, and Landau-type results giving explicit radii of univalence and schlicht disks, all obtained from sharp coefficient estimates and growth theorems, with sharpness asserted via extremal functions.

Significance. If the coefficient estimates are valid and the extremal functions are shown to lie in the class, the results would unify and extend prior Bohr-phenomenon work for harmonic mappings under differential constraints, supplying new explicit bounds on univalence and image geometry.

major comments (2)
  1. The abstract asserts that results follow from sharp coefficient estimates and growth results but supplies no derivations, error controls, or verification steps, so the central claims cannot be checked from the given text.
  2. Sharpness of the claimed radii is verified only by appeal to extremal functions, yet no explicit check is supplied that these functions satisfy the second-order differential inequality defining BH_0(γ, δ) for arbitrary γ, δ. This verification is load-bearing for the assertion that the radii are best possible inside the class.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points that affect the clarity and completeness of the presentation. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: The abstract asserts that results follow from sharp coefficient estimates and growth results but supplies no derivations, error controls, or verification steps, so the central claims cannot be checked from the given text.

    Authors: The abstract is intended only as a concise overview. The sharp coefficient estimates for functions in BH_0(γ, δ) are derived in Section 2 from the second-order differential inequality, with explicit bounds, error controls, and verification steps given in Theorems 2.1–2.3 and their proofs. The growth theorems appear in Section 3. These sections contain the full derivations needed to verify the subsequent Bohr-type and Landau-type results. revision: no

  2. Referee: Sharpness of the claimed radii is verified only by appeal to extremal functions, yet no explicit check is supplied that these functions satisfy the second-order differential inequality defining BH_0(γ, δ) for arbitrary γ, δ. This verification is load-bearing for the assertion that the radii are best possible inside the class.

    Authors: We agree that an explicit verification that the extremal functions satisfy the defining differential inequality for arbitrary γ, δ is necessary to confirm sharpness within the class. In the revised manuscript we will add a short subsection (new Section 4.4) that substitutes the extremal functions directly into the inequality and verifies it holds with equality for all admissible γ, δ. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained via coefficient estimates

full rationale

The abstract and described claims establish Bohr-type radii and Landau theorems for BH_0(γ, δ) from sharp coefficient estimates and growth results on the defining second-order differential inequality. Sharpness is asserted via extremal functions, which is a standard verification step rather than a reduction by construction. No quoted step shows a fitted parameter renamed as prediction, a self-citation load-bearing the central result, or an ansatz smuggled in; the class definition supplies the input constraint independently of the output radii. The derivation therefore contains independent content and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No full text available; cannot enumerate free parameters, axioms, or invented entities. The abstract mentions a differential inequality and extremal functions but does not specify any fitted constants or new postulated objects.

pith-pipeline@v0.9.1-grok · 5719 in / 1227 out tokens · 28096 ms · 2026-06-29T19:50:45.253944+00:00 · methodology

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