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arxiv: 2605.20089 · v1 · pith:KKZY67KGnew · submitted 2026-05-19 · 🧮 math.CV

Logarithmic Coefficients Problems of Geometric Subclass of Closed-to-convex Functions

Chayani Dhara , Nirupam Ghosh This is my paper

Pith reviewed 2026-05-20 03:08 UTC · model grok-4.3

classification 🧮 math.CV
keywords logarithmic coefficientsinverse logarithmic coefficientsHankel determinantclosed-to-convex functionsunit diskanalytic functionsreal-part conditionssharp bounds
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The pith

Sharp upper and lower bounds are derived for |γ₂| − |γ₁| and |Γ₂| − |Γ₁| in the class W(α) of analytic functions.

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 W(α) for α ≥ 0 as the normalized analytic functions f in the unit disk satisfying Re{f'(z) + α z f''(z)} > 0. It derives sharp bounds on the first few logarithmic coefficients γ_k and inverse logarithmic coefficients Γ_k for functions in this class. The main results give explicit sharp upper and lower estimates on the differences |γ₂| − |γ₁| and |Γ₂| − |Γ₁|, together with a sharp bound on the second Hankel determinant of these coefficients when α = 1.

Core claim

For f in W(α), the logarithmic coefficients satisfy sharp inequalities on |γ₂| − |γ₁| and |Γ₂| − |Γ₁|, with the bounds attained by explicitly identified extremal functions; when α = 1 the second Hankel determinant of the logarithmic coefficients also obeys a sharp inequality.

What carries the argument

The class W(α) defined by the condition Re{f'(z) + α z f''(z)} > 0, together with the logarithmic coefficients extracted from log(f(z)/z) = 2 ∑ γ_k z^k.

If this is right

  • The bounds hold with equality for specific functions that map the unit disk onto a slit plane or convex domain.
  • When α = 1 the second Hankel determinant |γ₁ γ₂; γ₂ γ₃| − |γ₂|² is bounded by a concrete constant.
  • The same technique yields corresponding estimates for the inverse logarithmic coefficients Γ_k.
  • The results extend the classical coefficient estimates for starlike and convex functions to this larger geometric subclass.

Where Pith is reading between the lines

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

  • These difference bounds may simplify estimates for the growth of partial sums of the logarithmic series.
  • The method could be adapted to obtain similar sharp differences for higher-order Hankel determinants in related classes.
  • Numerical verification for random points in the unit disk could confirm the sharpness for non-extremal functions.

Load-bearing premise

The extremal functions that attain the claimed sharp bounds belong to the class W(α) and can be identified explicitly.

What would settle it

Construct the candidate extremal function for a fixed α, compute its |γ₂| − |γ₁| explicitly, and check whether the value equals the stated bound.

read the original abstract

For $\alpha\ge 0$, let $\mathcal{W}(\alpha)$ be the class of all analytic functions in the unit disk $\mathbb{D}$ with normalization $f(0) = 0 $ and $ f'(0) = 1 $ that satisfy the relation $Re\,\{f'(z) + \alpha z f''(z)\} > 0$. This article aims to establish sharp bounds for logarithmic coefficients $\gamma_1$, $\gamma_2$ and $\gamma_3$ and logarithmic inverse coefficients $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ of functions in $\mathcal{W}(\alpha)$. The sharp upper and lower bounds for $\bigl|\,\gamma_2 \,\bigr|-\bigl|\,\gamma_1\,\bigr|$ and $\bigl|\,\Gamma_2 \,\bigr|-\bigl|\,\Gamma_1\,\bigr|$ have been obtained for the class $\mathcal{W}{(\alpha)}$. In addition, we establish sharp inequality for the second Hankel determinant of the logarithmic and inverse logarithmic coefficients for the class $\mathcal{W}{(1)}$.

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 defines the class W(α) for α ≥ 0 consisting of normalized analytic functions f in the unit disk satisfying Re{f'(z) + α z f''(z)} > 0. It derives sharp upper and lower bounds on |γ₂| − |γ₁| and |Γ₂| − |Γ₁| for the logarithmic coefficients γ_k and inverse logarithmic coefficients Γ_k of functions in W(α), together with a sharp bound on the second Hankel determinant of these coefficients when α = 1.

Significance. If the sharpness assertions are verified, the results extend classical coefficient estimates to a geometrically defined subclass related to closed-to-convex functions, supplying explicit bounds that may serve as benchmarks for further work on logarithmic coefficients in geometric function theory. The reliance on standard relations among coefficients rather than ad-hoc fitting is a methodological strength.

major comments (2)
  1. [Sharpness statements for |γ₂|−|γ₁| and |Γ₂|−|Γ₁|] The sharpness claims for |γ₂| − |γ₁| and |Γ₂| − |Γ₁| (stated after the main coefficient theorems) rest on the existence of an extremal function f_α ∈ W(α) for every α ≥ 0. The candidate obtained by solving f' + α z f'' = (1 + z)/(1 − z) must be shown to satisfy the defining real-part inequality on a set of positive measure; if it fails for some α > 0, the stated bounds reduce to non-sharp estimates.
  2. [Hankel-determinant result for α = 1] For the sharp Hankel-determinant inequality when α = 1, the extremal function must be exhibited explicitly and verified to lie in W(1); the current argument appears to invoke subordination or variational methods without confirming membership for the full range of parameters.
minor comments (2)
  1. [Throughout] Notation for the class should be uniformly rendered as script W(α) in all statements and proofs.
  2. [Preliminaries] A brief remark on how the logarithmic coefficients are extracted from the series expansion of log(f(z)/z) would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major comment below and indicate the revisions we plan to make.

read point-by-point responses

  1. Referee: The sharpness claims for |γ₂| − |γ₁| and |Γ₂| − |Γ₁| (stated after the main coefficient theorems) rest on the existence of an extremal function f_α ∈ W(α) for every α ≥ 0. The candidate obtained by solving f' + α z f'' = (1 + z)/(1 − z) must be shown to satisfy the defining real-part inequality on a set of positive measure; if it fails for some α > 0, the stated bounds reduce to non-sharp estimates.

    Authors: We appreciate the referee pointing this out. By the very definition of the extremal function, it satisfies the differential equation f'(z) + α z f''(z) = (1 + z)/(1 - z) for |z| < 1. The function (1 + z)/(1 - z) has positive real part in the unit disk, specifically Re{(1 + z)/(1 - z)} = (1 - |z|^2)/|1 - z|^2 > 0. Therefore, Re{f'(z) + α z f''(z)} > 0 holds for all z in D, which means f_α belongs to W(α) for every α ≥ 0. This establishes the sharpness. We will include a short explanatory sentence in the revised version to make this explicit. revision: yes

  2. Referee: For the sharp Hankel-determinant inequality when α = 1, the extremal function must be exhibited explicitly and verified to lie in W(1); the current argument appears to invoke subordination or variational methods without confirming membership for the full range of parameters.

    Authors: Thank you for this suggestion. For α = 1, the extremal function is the unique solution to f' + z f'' = (1 + z)/(1 - z), which, as noted above, satisfies the membership condition Re{f' + z f''} > 0 by construction. We will revise the relevant section to explicitly identify this function and confirm its inclusion in W(1), thereby clarifying the sharpness of the Hankel determinant bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity in coefficient bound derivations for class W(α)

full rationale

The paper defines the class W(α) directly via the real-part condition Re{f'(z) + α z f''(z)} > 0 with standard normalization, then derives bounds on logarithmic coefficients γ_k and inverse coefficients Γ_k using classical coefficient relations, subordination principles, and extremal function identification within the class. No steps reduce by construction to fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations whose validity depends on the present results. Sharpness assertions rest on verifying that candidate functions satisfy the defining inequality of W(α), which is an independent check rather than a tautology. The derivation chain is self-contained against external benchmarks in univalent function theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the definition of the class W(α) together with standard facts from geometric function theory about coefficient relations and subordination. No new entities are postulated and no numerical parameters appear to be fitted inside the abstract.

axioms (2)
  • domain assumption The functions are analytic and normalized by f(0)=0, f'(0)=1 inside the unit disk.
    This is the standard normalization invoked in the abstract for the class definition.
  • domain assumption The real-part inequality Re{f'(z) + α z f''(z)} > 0 holds for all z in the unit disk.
    This is the defining property of W(α) stated in the abstract.

pith-pipeline@v0.9.0 · 5731 in / 1453 out tokens · 44183 ms · 2026-05-20T03:08:49.434909+00:00 · methodology

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Reference graph

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