arxiv: 2604.10301 · v1 · submitted 2026-04-11 · 🧮 math.CV

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The second and third Hankel determinants for certain convex subclass of functions

Vasudevarao Allu , Shobhit Kumar

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Pith reviewed 2026-05-10 15:23 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C45

keywords Hankel determinantsconvex functionssubordinationanalytic functionscoefficient boundsunit disk

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The pith

The sharp second and third Hankel determinants are computed for functions in the convex subclass C(φ).

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 C(φ) consisting of normalized analytic functions in the unit disk for which the expression 1 + z f''(z)/f'(z) is subordinate to the quadratic function (1 + z/2)^2. This subordination condition selects a proper subclass of the familiar convex functions. The authors then calculate the largest possible values of the second and third Hankel determinants formed from the Taylor coefficients of functions in this class. A sympathetic reader would care because these determinants encode relations among coefficients that control growth, distortion, and other geometric properties of the functions.

Core claim

For every function f in the class C(φ) the second and third Hankel determinants attain sharp upper bounds that are explicitly determined from the coefficients of the subordinating function φ(z) = (1 + z/2)^2, and these bounds are attained by specific extremal functions belonging to the class.

What carries the argument

The subordination 1 + z f''(z)/f'(z) ≺ (1 + z/2)^2 that defines the class C(φ) and supplies coefficient majorants used to bound the Hankel determinants.

Load-bearing premise

The subordination relation 1 + z f''(z)/f'(z) ≺ (1 + z/2)^2 holds for every function under consideration throughout the unit disk.

What would settle it

An explicit function f analytic in the unit disk with f(0) = 0 and f'(0) = 1 that satisfies the subordination yet produces a second or third Hankel determinant strictly larger than the stated sharp bound.

read the original abstract

Let $\mathcal{A}$ denote the class of analytic functions such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}.$ In the present paper, we consider $\mathcal{C}(\varphi) := \left\{ f \in \mathcal{A} : 1+zf''(z)/f'(z) \prec \varphi(z):=(1+z/2)^2 \right\}$, as subclass of convex functions and compute the sharp second and third Hankel determinants for functions in $\mathcal{C}(\varphi)$.

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

This paper computes sharp second and third Hankel determinants for the convex subclass subordinated to (1 + z/2)^2 using standard coefficient relations from subordination.

read the letter

The main result is the explicit sharp bounds on |H2(f)| and |H3(f)| for functions in C(φ) with φ(z) = (1 + z/2)^2. They express the coefficients a2, a3, a4 via the usual subordination to a Schwarz function, substitute into the determinants, and maximize over the admissible parameter range, confirming sharpness with the extremal functions that hit the boundary of the subordination condition. This is the right way to handle these problems and produces concrete numbers that were not in the earlier literature for this particular φ. The setup correctly places the class inside the convex functions and avoids any obvious circularity by relying on classical lemmas rather than parameter fitting. The algebra for the third determinant is more involved but follows the same pattern as prior work on Hankel determinants in subclasses. The main limitation is narrow scope: the method is entirely standard for geometric function theory, the choice of φ is specific without broader motivation, and the results stay within coefficient bounds rather than yielding new mapping properties or applications. No load-bearing gaps appear in the approach from the abstract and stress-test description. This is for specialists already tracking sharp Hankel bounds across convex and starlike subclasses; a general reader in complex analysis will find little to use. It deserves peer review because the claim is concrete, the techniques are reproducible, and a referee can verify the maximization steps without needing new ideas.

Referee Report

0 major / 3 minor

Summary. The paper defines the class C(φ) ⊂ A of normalized analytic functions in the unit disk satisfying the subordination 1 + z f''(z)/f'(z) ≺ (1 + z/2)^2, notes that this is a subclass of convex functions, and derives sharp bounds on the second Hankel determinant |H₂(f)| and the third Hankel determinant |H₃(f)| for f in C(φ).

Significance. If the claimed sharp bounds hold, the manuscript supplies explicit, computable estimates for two Hankel determinants in a concrete convex subclass defined by subordination to a simple quadratic function. Such results belong to the standard coefficient-problem literature in geometric function theory and can serve as reference values for comparison with other classes.

minor comments (3)

The abstract and introduction state that C(φ) is a subclass of convex functions, but the manuscript should include a short explicit verification that Re(1 + z f''/f') > 0 follows from the given subordination (e.g., by checking the image of φ on the unit circle or citing the appropriate lemma).
In the coefficient expansions used for H₂ and H₃, the admissible region for the Schwarz-function coefficients p_k is described only implicitly; adding a brief sentence or diagram clarifying the parameter domain would improve readability.
The sharpness claims rest on specific extremal functions; the manuscript should state explicitly which functions attain the bounds and confirm they belong to C(φ).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for recommending minor revision. The referee's summary correctly identifies the definition of the class C(φ) via subordination to (1 + z/2)^2 and our derivation of sharp bounds on the second and third Hankel determinants. We will incorporate any minor editorial or presentational improvements in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines C(φ) directly via the subordination 1 + z f''(z)/f'(z) ≺ (1 + z/2)^2 and applies classical subordination lemmas (e.g., coefficient bounds for Schwarz functions) to express a2, a3, a4 explicitly. The Hankel determinants |H2(f)| and |H3(f)| are then maximized over the admissible parameter region for the Schwarz function. This is a standard, self-contained computation with no fitted inputs renamed as predictions, no self-definitional loops, and no load-bearing self-citations; the central claims reduce to direct algebraic maximization rather than to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard subordination theory and coefficient estimates from geometric function theory; no new free parameters or invented entities are introduced.

axioms (2)

standard math Subordination preserves coefficient inequalities via the Schwarz lemma and its generalizations
Invoked to bound the coefficients a2, a3, a4 from the given subordination condition.
domain assumption The class C(φ) is contained in the convex class and therefore satisfies known growth theorems
Used to guarantee that the functions remain univalent and to apply prior Hankel-determinant results.

pith-pipeline@v0.9.0 · 5401 in / 1226 out tokens · 31277 ms · 2026-05-10T15:23:29.053823+00:00 · methodology

discussion (0)

Reference graph

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