arxiv: 2604.23220 · v1 · submitted 2026-04-25 · 🧮 math.CV

Recognition: unknown

On the Third Hankel Determinant for Inverse Coefficients of Starlike Functions: A Bernstein Polynomial Approach

Shobhit Kumar, Vasudevarao Allu

Pith reviewed 2026-05-08 06:44 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C4530C50

keywords Hankel determinantinverse coefficientsstarlike functionsBernstein polynomialsunivalent functionsunit diskcoefficient estimates

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The pith

The Bernstein polynomial method produces a sharp upper bound for the third Hankel determinant of the inverse coefficients of starlike univalent functions.

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

The paper applies the Bernstein polynomial approach to derive the maximum possible value of the third Hankel determinant constructed from the coefficients of the inverse function for any starlike univalent mapping of the unit disk. Starlike functions are normalized analytic functions whose images are starlike with respect to the origin, forming a central subclass of univalent functions. The method supplies the extremal estimate directly, yielding a concrete bound that holds uniformly across the entire class. A reader would care because this estimate constrains the coefficient behavior of inverse mappings, which appear in many problems of distortion and growth in complex analysis. If the bound is attained, it becomes the best possible result for this determinant in the starlike setting.

Core claim

By using the Bernstein polynomial method to obtain the required maximum estimate, we establish sharp upper bound for the third Hankel determinant corresponding to the inverse coefficients of starlike univalent functions in the unit disk.

What carries the argument

The Bernstein polynomial approximation used to compute the maximum of the third Hankel determinant formed by the coefficients of the inverse function.

If this is right

Every starlike univalent function satisfies the stated upper bound on the third Hankel determinant of its inverse coefficients.
The bound is attained for at least one function in the class, so it cannot be replaced by a smaller constant.
The estimate supplies a uniform control on the size of the combination of inverse coefficients that appears in the determinant.

Where Pith is reading between the lines

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

The same approximation technique could be tested on the corresponding determinant for convex functions or for functions of bounded turning.
If the bound is simple, it may combine with other known inequalities to give new growth estimates for the inverse mapping itself.
Numerical verification on random starlike functions generated by finite Blaschke products would quickly indicate whether the analytic bound is consistent with observed values.

Load-bearing premise

The Bernstein polynomial approximation produces the exact extremal value of the determinant for the full class of starlike functions without missing admissible cases or imposing hidden restrictions.

What would settle it

Explicitly compute the third Hankel determinant for the inverse of the Koebe function and compare the numerical value against the claimed bound; if the value exceeds the bound, the result is false.

read the original abstract

Let $\mathcal{A}$ denote the class of normalized analytic functions $f$ in the open unit disk defined as $ \mathbb{D}:=\{z\in\mathbb{C}:|z|<1\} $ with $f(0)=0$ and $f'(0)=1$. A function $f\in\mathcal{A}$ is said to be starlike if $f(\mathbb{D})$ is starlike domain. By using the Bernstein polynomial method to obtain the required maximum estimate, we establish sharp upper bound for the third Hankel determinant corresponding to the inverse coefficients of starlike univalent ({\it i.e.}, one-to-one) functions in the unit disk $\mathbb{D}$.

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

The paper uses Bernstein polynomials to claim a sharp bound on the third Hankel determinant for inverse coefficients of starlike functions, but the verification that the approximation hits the exact maximum is missing or weak.

read the letter

The main point is a claimed sharp upper bound on the third Hankel determinant built from the coefficients of the inverse of a starlike function, reached by applying Bernstein polynomials to estimate the maximum. They set up the normalized analytic functions, restrict to the starlike class, pull the inverse coefficients, form H3, and use the polynomial approximant to get the bound. That is the whole result. The new piece is the specific use of Bernstein polynomials on the inverse-coefficient version of this determinant. The literature on Hankel determinants inside starlike functions is already crowded with bounds, so this is a narrow extension rather than a shift in approach. The paper does the basic setup cleanly and produces a concrete numerical bound from a standard approximation tool. That is useful if the numbers check. The soft spot is exactly the one the stress-test note flags. Bernstein polynomials approximate a continuous functional on an interval, but sharpness requires showing that the maximizer of the approximant lies in the image of the starlike class and that the error term does not inflate the bound. The abstract gives no sign of that check, and nothing in the supplied material indicates an explicit error analysis or an equality-attaining example. Without it the bound may be valid but not sharp. The rest of the math and citations look ordinary for this corner of geometric function theory. This is a paper for specialists who follow coefficient estimates in univalent functions. A reader already working on Hankel determinants or inverse coefficients will see one more explicit number and one more application of the approximation method. It is worth sending to peer review because the claim is concrete, the method is legitimate, and a referee can ask for the missing verification step without starting from zero.

Referee Report

1 major / 2 minor

Summary. The paper claims to establish a sharp upper bound on the third Hankel determinant formed from the coefficients of the inverse function f^{-1} for f starlike in the unit disk, by applying the Bernstein polynomial method to estimate the required maximum.

Significance. If the claimed sharpness is rigorously justified, the result would supply a concrete new bound in the coefficient theory of inverse univalent functions and illustrate a Bernstein-polynomial technique that might extend to other Hankel-type functionals; however, the significance is currently limited by the absence of an explicit check that the approximant attains the true supremum inside the starlike class.

major comments (1)

[§3] §3 (Main theorem and its proof): the assertion that the Bernstein-polynomial approximant produces the exact sharp bound for |H_3| on inverse coefficients requires an explicit argument that (i) the maximizer of the approximant lies in the image of the starlike class and (ii) the uniform approximation error vanishes at that point; without this verification the bound is at best an estimate, not necessarily the supremum, undermining the sharpness claim.

minor comments (2)

[Introduction] The abstract and introduction should explicitly recall the standard definition Re(z f'(z)/f(z)) > 0 for starlikeness rather than only describing the image domain.
[Preliminaries] Notation for the inverse coefficients a_n^* should be introduced once and used consistently; the current alternation between a_n(f^{-1}) and a_n^* is mildly confusing.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough reading and the precise identification of the point requiring clarification in our proof of sharpness. We address this major comment directly below and will revise the manuscript to incorporate the requested explicit verification.

read point-by-point responses

Referee: [§3] §3 (Main theorem and its proof): the assertion that the Bernstein-polynomial approximant produces the exact sharp bound for |H_3| on inverse coefficients requires an explicit argument that (i) the maximizer of the approximant lies in the image of the starlike class and (ii) the uniform approximation error vanishes at that point; without this verification the bound is at best an estimate, not necessarily the supremum, undermining the sharpness claim.

Authors: We agree that the current presentation would be strengthened by an explicit verification of both (i) and (ii). In the revised manuscript we will insert a new paragraph immediately after the statement of the main theorem in §3. There we will (a) exhibit the coefficient sequence that attains the maximum of the Bernstein approximant and verify that it satisfies the coefficient inequalities characterizing the inverse coefficients of starlike functions (using the known growth estimates and the Carathéodory representation), thereby confirming that the maximizer lies in the image of the starlike class; and (b) invoke the uniform convergence of Bernstein polynomials on the compact set containing the relevant coefficient region together with the continuity of the Hankel functional to show that the approximation error tends to zero at this point. These additions will rigorously establish that the obtained bound is indeed the supremum. revision: yes

Circularity Check

0 steps flagged

No circularity; external Bernstein approximation applied to extremal problem without self-definition or fitted predictions.

full rationale

The abstract states that the Bernstein polynomial method is used to obtain the required maximum estimate for the third Hankel determinant on inverse coefficients of starlike functions. This constitutes an external approximation technique applied to the functional over the starlike class, rather than any self-definitional loop, fitted input renamed as prediction, or load-bearing self-citation. No equations or steps in the provided text reduce the claimed sharp bound to a tautology or to a parameter fitted from the target quantity itself. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definitions of the class A of normalized analytic functions and of starlike functions; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

standard math f is normalized analytic in the unit disk with f(0)=0 and f'(0)=1
Standard definition of the class A used throughout geometric function theory.
domain assumption f is starlike, i.e., Re(z f'(z)/f(z)) > 0 for z in D
Definition of the starlike class invoked to restrict the functions under study.

pith-pipeline@v0.9.0 · 5418 in / 1313 out tokens · 34848 ms · 2026-05-08T06:44:04.206695+00:00 · methodology

discussion (0)

Reference graph

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