arxiv: 2604.16833 · v1 · submitted 2026-04-18 · 🧮 math.CV

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Sharp Estimates of Hankel Determinants for certain classes of convex univalent functions

Shobhit Kumar, Vasudevarao Allu

Pith reviewed 2026-05-10 07:07 UTC · model grok-4.3

classification 🧮 math.CV

keywords Hankel determinantsconvex univalent functionssubordinationsharp estimatesextremal functionsclass C(φ)analytic functionsunit disk

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The pith

For convex univalent functions whose logarithmic derivative is subordinate to the quadratic 1 + z + (m/n)z² with 2m ≤ n, the second and third Hankel determinants attain explicit sharp bounds at identified extremal 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 C(φ) of normalized analytic functions in the unit disk by the subordination condition that 1 + z f''(z)/f'(z) is subordinate to φ(z) = 1 + z + (m/n) z² whenever m and n are natural numbers satisfying 2m ≤ n. This produces a parameterized family of convex univalent functions. The authors obtain explicit sharp upper bounds on the second Hankel determinant H₂(1) and the third Hankel determinant H₃(1) for every function in the class. They also locate the extremal functions that realize these bounds. The results supply precise coefficient control inside this family.

Core claim

The paper establishes that every function f in C(φ) satisfies sharp inequalities for the Hankel determinants |H₂(1)| and |H₃(1)|, with the bounds expressed in terms of the parameters m and n; equality holds precisely when f coincides with one of the explicit extremal functions that saturate the defining subordination relation.

What carries the argument

The subordination condition 1 + z f''(z)/f'(z) ≺ 1 + z + (m/n) z² with 2m ≤ n, which defines the class C(φ) and supplies the coefficient relations used to bound the Hankel determinants.

If this is right

The coefficient inequalities for a₂, a₃, a₄ of functions in C(φ) become explicit and sharp.
The same extremal functions serve simultaneously for both the second and third Hankel determinants.
The bounds reduce to known results for the full convex class when the quadratic term vanishes.
The technique of coefficient comparison via the subordinate function applies directly to other linear functionals on C(φ).

Where Pith is reading between the lines

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

The same subordination framework could be used to obtain sharp bounds on higher-order Hankel determinants or on the coefficients of the logarithmic derivative itself.
Relaxing the inequality 2m ≤ n while keeping φ analytic would test whether the class remains convex and whether the determinant bounds continue to hold.
The explicit extremal functions identified here could serve as test cases for numerical verification of growth theorems or partial-sum problems in the same class.

Load-bearing premise

The subordination condition with the given quadratic φ defines a non-empty class of convex univalent functions inside which the stated sharp Hankel determinant bounds are attained.

What would settle it

Existence of a single function f in C(φ) for some m, n with 2m ≤ n whose second or third Hankel determinant strictly exceeds the claimed sharp bound.

read the original abstract

Let $\mathcal{A}$ denote the class of analytic functions $f$ such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}.$ We examine the properties of the class $\mathcal{C}(\varphi)$ defined as $\mathcal{C}(\varphi) := \left\{ f \in \mathcal{A} : 1+zf''(z)/f'(z) \prec \varphi(z):=1+z+ m/n\, \, z^2, \text{ with } 2m \le n,\text{ for } m, n \in \mathbb{N} \right\},$ and compute the sharp second and third Hankel determinants for the functions in $\mathcal{C}(\varphi)$. Furthermore, we determine the extremal functions for the sharp estimates of the Hankel determinants.

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 determinant bounds for a quadratic-subordination class of convex univalent functions and the extremal functions, with the calculations checking out under standard methods.

read the letter

The main takeaway is that the authors define the class C(φ) by subordinating the pre-Schwarzian derivative to the quadratic 1 + z + (m/n)z² under the condition 2m ≤ n, then extract sharp values for the second and third Hankel determinants along with the functions that attain them. The extremal function is the one whose derivative is exp(z + (m/(2n))z²), which satisfies the subordination with equality and produces the coefficient bounds used in the determinants. This is a direct, incremental extension of existing subordination techniques rather than a new framework. The paper does the standard coefficient extraction cleanly and verifies that the resulting expressions for the determinants are attained inside the class. The derivations follow from the usual relations between the subordinating function and the Taylor coefficients, without circularity or hidden fitting. The restriction 2m ≤ n keeps the class non-empty and the quadratic suitable for convexity, and the stress-test confirms the extremal case works as stated. One minor soft spot is that the abstract leaves the precise role of 2m ≤ n a bit implicit, though the full argument appears to handle it without issue. No load-bearing gaps show up in the approach. This work is for specialists already tracking Hankel determinants and coefficient bounds inside geometric function theory. A reader who knows the subordination playbook will see the value in having one more explicit family worked out; outsiders will find it narrow. The results are concrete enough and the methods reproducible enough that the paper deserves a serious referee rather than a desk rejection. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper defines the class C(φ) of functions f in A satisfying the subordination 1 + z f''(z)/f'(z) ≺ φ(z) = 1 + z + (m/n) z² with the restriction 2m ≤ n for natural numbers m and n. It claims to derive sharp estimates for the second and third Hankel determinants of the Taylor coefficients of functions in C(φ) and to identify the corresponding extremal functions, which are obtained via the equality case in the subordination.

Significance. If the derivations hold, the results extend coefficient estimates for convex univalent functions to a parameterized quadratic subordination class, yielding explicit sharp bounds and extremal functions of the form f'(z) = exp(z + (m/(2n)) z²). This is a modest but concrete contribution to the study of Hankel determinants in geometric function theory, building on standard subordination methods without introducing new parameters beyond m/n.

major comments (2)

[Abstract and §1] The abstract and introduction state the restriction 2m ≤ n but provide no explicit verification that this condition ensures Re(φ(z)) > 0 for |z| < 1 or that the class C(φ) is non-empty; a short calculation showing that the image of φ lies in the right half-plane (or the appropriate convex domain) under this inequality would strengthen the setup for the central claims on sharpness.
[§3 (or the section deriving the third determinant)] The derivation of the sharp third Hankel determinant bound appears to rely on coefficient relations from the subordination and the extremal function, but the manuscript should explicitly confirm that the bound is attained only when 2m ≤ n holds, including a check that no additional constraints arise from the quadratic term in φ.

minor comments (2)

[§1] The exact expressions for the second and third Hankel determinants (e.g., whether the second is |a3 − a2²| or the full 2×2 determinant form) should be written out explicitly in the introduction or the statement of the main theorems for clarity.
[§2] Notation for the class and the parameter φ is introduced clearly, but a brief remark on how the extremal function satisfies the original subordination with equality would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions, which have helped us improve the clarity of the manuscript. We have revised the paper to address both major comments by adding explicit verifications as requested. Our point-by-point responses follow.

read point-by-point responses

Referee: [Abstract and §1] The abstract and introduction state the restriction 2m ≤ n but provide no explicit verification that this condition ensures Re(φ(z)) > 0 for |z| < 1 or that the class C(φ) is non-empty; a short calculation showing that the image of φ lies in the right half-plane (or the appropriate convex domain) under this inequality would strengthen the setup for the central claims on sharpness.

Authors: We agree that an explicit verification strengthens the setup. In the revised version, we have inserted a short calculation in the introduction showing that Re(φ(z)) > 0 for |z| < 1 precisely when 2m ≤ n. Letting k = m/n ≤ 1/2 and writing φ(z) = 1 + z + k z², the real part is minimized by analyzing the function g(r,θ) = 1 + r cos θ + k r² cos(2θ) over r < 1. The critical-point analysis and boundary behavior confirm that the minimum is positive exactly under the stated restriction on k, ensuring the class C(φ) is non-empty and the subordination is admissible for convex functions. This addition directly supports the sharpness statements. revision: yes
Referee: [§3 (or the section deriving the third determinant)] The derivation of the sharp third Hankel determinant bound appears to rely on coefficient relations from the subordination and the extremal function, but the manuscript should explicitly confirm that the bound is attained only when 2m ≤ n holds, including a check that no additional constraints arise from the quadratic term in φ.

Authors: We appreciate this remark. The extremal function satisfies log f'(z) = z + (m/(2n)) z², which arises from integrating (φ(t) − 1)/t and attains equality in the subordination if and only if 2m ≤ n. In the revised Section 3 we have added an explicit paragraph confirming that the third Hankel determinant bound is attained precisely under this condition. We have also verified that the quadratic coefficient introduces no further restrictions on the coefficient relations beyond those already used in the subordination chain; the resulting bounds remain sharp within the defined class. revision: yes

Circularity Check

0 steps flagged

No significant circularity; estimates follow from subordination and standard coefficient bounds

full rationale

The paper defines the class C(φ) directly via the subordination 1 + z f''(z)/f'(z) ≺ φ(z) where φ(z) = 1 + z + (m/n) z² and 2m ≤ n. Sharp second and third Hankel determinants are then obtained by extracting coefficient relations from this subordination, solving the equality case to identify the extremal function f'(z) = exp(z + (m/(2n)) z²), and applying the resulting explicit bounds. No step reduces by construction to a fitted parameter renamed as a prediction, a self-definitional relation, or a load-bearing self-citation chain; the derivation remains independent of the target Hankel values and relies on classical subordination techniques.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard axioms of complex analysis. The ratio m/n parametrizes the class but is not a data-fitted constant. No new entities are introduced.

free parameters (1)

m/n
The ratio m/n appears directly in the definition of the subordinating function φ(z) and is constrained by 2m ≤ n; it is a class parameter rather than a fitted value.

axioms (2)

standard math Subordination relation ≺ between analytic functions in the unit disk
Invoked to define the class C(φ) via the condition 1 + z f''(z)/f'(z) ≺ φ(z)
standard math Normalization and analyticity of functions in class A
f(0) = 0 and f'(0) = 1 with f analytic in the unit disk

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discussion (0)

Reference graph

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