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arxiv: 2605.18087 · v1 · pith:AHENZQ6Mnew · submitted 2026-05-18 · 🧮 math.CA

Sharp Estimates for Conjugate Functions with Applications to Trigonometric Polynomials

Silouanos Brazitikos This is my paper

Pith reviewed 2026-05-20 00:22 UTC · model grok-4.3

classification 🧮 math.CA
keywords conjugate functionsharmonic majoranthalf-striptrigonometric polynomialssharp estimateslogarithmic lossminima of polynomials
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The pith

A harmonic majorant in a half-strip yields a sharp estimate for conjugate functions and removes the logarithmic loss from bounds on minima of trigonometric polynomials.

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

The paper establishes a sharp pointwise estimate for conjugate functions by constructing a harmonic majorant inside a half-strip. This construction supplies the optimal constant without the extra factors that appeared in earlier work. The same estimate is then applied to trigonometric polynomials, removing the logarithmic loss present in Papadopoulos' theorem and delivering the best possible order for the lower bound on their minima. A reader would care because conjugate-function bounds are basic tools in harmonic analysis, and tightening them produces cleaner inequalities for polynomials that arise in approximation and Fourier analysis.

Core claim

We prove a sharp estimate for conjugate functions using a harmonic majorant in a half-strip. As an application, we remove the logarithmic loss from a theorem of Papadopoulos on minima of trigonometric polynomials and obtain the optimal order in the corresponding inequality.

What carries the argument

The harmonic majorant constructed in the half-strip domain, which supplies the sharp constant for the conjugate-function estimate.

Load-bearing premise

The harmonic majorant constructed in the half-strip domain yields the sharp constant for the conjugate function estimate without additional losses.

What would settle it

An explicit numerical computation for a low-degree trigonometric polynomial that checks whether the minimum attains the predicted order without any logarithmic factor.

read the original abstract

We prove a sharp estimate for conjugate functions using a harmonic majorant in a half-strip. As an application, we remove the logarithmic loss from a theorem of Papadopoulos on minima of trigonometric polynomials and obtain the optimal order in the corresponding inequality.

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 / 2 minor

Summary. The manuscript proves a sharp estimate for conjugate functions of harmonic functions by constructing a specific harmonic majorant in a half-strip domain. As an application, the estimate is used to remove the logarithmic loss from Papadopoulos' theorem on the minima of trigonometric polynomials, yielding the optimal order in the corresponding inequality.

Significance. If the central sharpness claim holds, the work would strengthen sharp constant results in harmonic analysis via direct analytic methods with a harmonic majorant, avoiding parameter fitting or circular reductions. The application to trigonometric polynomial minima would improve upon prior bounds by achieving the optimal order without extraneous logarithmic factors, which is a concrete advance for extremal problems in Fourier analysis.

major comments (2)
  1. [Main estimate section (half-strip majorant construction)] The section constructing the harmonic majorant in the half-strip (around the main estimate for the conjugate function): it is not shown that this particular majorant is minimal or that equality is attained in a limiting case. Without an explicit comparison to the least harmonic majorant or a boundary function achieving the bound, the claimed sharpness of the constant for the conjugate estimate remains unverified and could contain a hidden factor.
  2. [Application to trigonometric polynomials] The application section transferring the estimate to Papadopoulos' theorem on minima of trigonometric polynomials: the removal of the logarithmic loss and attainment of optimal order relies directly on the conjugate estimate being sharp with no multiplicative loss. If the majorant introduces any constant >1, the claimed optimal order would not follow.
minor comments (2)
  1. [Abstract] The abstract refers to a 'half-strip' without specifying the exact width or the boundary data used for the majorant; adding these details would clarify the setup for readers.
  2. [Introduction] The introduction should include a precise statement of the logarithmic loss in the cited Papadopoulos theorem to make the improvement explicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's thorough review and recommendation for major revision. We have carefully considered the comments regarding the verification of sharpness in the harmonic majorant construction and its implications for the application. Below we provide point-by-point responses, and we have revised the manuscript to include an explicit demonstration of the limiting case where the bound is attained.

read point-by-point responses

  1. Referee: The section constructing the harmonic majorant in the half-strip (around the main estimate for the conjugate function): it is not shown that this particular majorant is minimal or that equality is attained in a limiting case. Without an explicit comparison to the least harmonic majorant or a boundary function achieving the bound, the claimed sharpness of the constant for the conjugate estimate remains unverified and could contain a hidden factor.

    Authors: We appreciate this observation. The harmonic majorant in the half-strip is constructed by taking the solution to the Dirichlet problem with boundary data chosen to be the extremal configuration for the conjugate function estimate. To verify minimality, we note that this majorant coincides with the Poisson integral of the boundary values that correspond to the case where the conjugate achieves its maximum ratio. We have added a paragraph in the revised manuscript demonstrating that equality is approached in the limiting case as the imaginary part tends to the boundary and for a specific family of functions approximating the sign function or the extremal harmonic function. This shows that the constant is sharp and no hidden multiplicative factor greater than one is introduced. revision: yes

  2. Referee: The application section transferring the estimate to Papadopoulos' theorem on minima of trigonometric polynomials: the removal of the logarithmic loss and attainment of optimal order relies directly on the conjugate estimate being sharp with no multiplicative loss. If the majorant introduces any constant >1, the claimed optimal order would not follow.

    Authors: We agree that the optimality in the application to trigonometric polynomials hinges on the conjugate estimate having no extra constant factor. As explained in our response to the previous comment, the revised version now explicitly confirms that the majorant yields the sharp constant. Consequently, the transfer of the estimate to Papadopoulos' result proceeds directly, removing the logarithmic loss and achieving the optimal order without any extraneous factors. We have added a brief remark in the application section referencing the sharpness verification. revision: partial

Circularity Check

0 steps flagged

No significant circularity; direct analytic construction

full rationale

The paper claims a sharp conjugate-function estimate obtained from an explicitly constructed harmonic majorant in a half-strip, then applies the estimate to remove a logarithmic loss in a prior result of Papadopoulos. No equations or steps are shown that reduce the claimed sharp constant to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose own justification is internal to the present work. The derivation chain therefore remains self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of harmonic functions and majorants in strip domains from complex analysis; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • standard math Existence and properties of harmonic majorants in half-strip domains for controlling conjugate functions
    Invoked to derive the sharp estimate; standard background in harmonic analysis.

pith-pipeline@v0.9.0 · 5547 in / 1103 out tokens · 40195 ms · 2026-05-20T00:22:56.115895+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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