An improvement of the estimates of the modulus of the Hankel determinants of second and third order for the class $\mathcal{S}$ of univalent functions

Milutin Obradovi\'c; Nikola Tuneski

REVIEW 3 major objections 5 minor 1 cited by

An improvement of the estimates of the modulus of the Hankel determinants of second and third order for the class $\mathcal{S}$ of univalent functions

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For univalent functions, this paper establishes the improved Hankel bounds $|H_2(2)| \leq 1.3614$ and $|H_3(1)| \leq 1.6787$.

desk verdict A genuine improvement in Hankel bounds via Grunsky coefficients, held back by an unverified numerical maximization step that should be certified before the H3 bound is taken as proven. read the letter →

arxiv 2411.12378 v1 pith:L5XKIS4Y submitted 2024-11-19 math.CV

Milutin Obradović , Nikola Tuneski This is my paper · ORCID

classification math.CV MSC 30C4530C5030C55

keywords univalentfunctionsHankeldeterminantsseconddeterminantthirdGrunskycoefficientsinequalityupperboundscoefficientestimates

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes improved upper bounds on the magnitudes of the second- and third-order Hankel determinants for the class of univalent (one-to-one) analytic functions on the unit disk: $|H_2(2)| \leq 1.3614\ldots$ and $|H_3(1)| \leq 1.6787\ldots$. These numbers replace the earlier bounds $3.666\ldots$ and $3.2588\ldots$ from [7], so the improvement is substantial. The proof rewrites the Taylor coefficients $a_2,\ldots,a_5$ using Grunsky coefficients and Grunsky’s inequality, then reduces each determinant to the maximum of an explicit two-variable function over a triangular domain. The final step, finding those maxima, is done by solving the critical-point systems numerically and checking the boundary.

What carries the argument

The machinery is the Grunsky coefficient system: the logarithmic transform $\log\frac{f(t)-f(z)}{t-z}$ for a univalent $f$ expands as $\sum_{p,q=0}^{\infty}\omega_{p,q}t^p z^q$, with symmetry $\omega_{p,q}=\omega_{q,p}$ and a constraint called Grunsky’s inequality. Working with the auxiliary function $f_2(z)=\sqrt{f(z^2)}$ makes only odd-indexed Grunsky coefficients appear, and relations from [3] express $a_2,\ldots,a_5$ as polynomials in these coefficients. Specializing Grunsky’s inequality gives the estimates (9) on $|\omega_{11}|,|\omega_{13}|,|\omega_{15}|,|\omega_{17}|$. These estimates convert the Hankel determinants into the explicit two-variable functions $F_1$ and $F_2$, whose maxima over the domain $D_1$ are then evaluated.

What would settle it

Take the domain $D_1$ and compute the global maxima of $F_1$ and $F_2$ with rigorous interval arithmetic; if either maximum exceeds $1.3614\ldots$ or $1.6787\ldots$, Theorem 2 is false. Equivalently, a concrete univalent function with $|H_2(2)|>1.3614\ldots$ or $|H_3(1)|>1.6787\ldots$ would refute the bounds.

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Extended reading notes

Core claim

The central claim is Theorem 2: for every normalized univalent function $f(z)=z+a_2z^2+a_3z^3+\cdots$, the quantities $|a_2 a_4 - a_3^2|$ and $|a_3(a_2 a_4 - a_3^2) - a_4(a_4 - a_2 a_3) + a_5(a_3 - a_2^2)|$ are bounded by $1.3614\ldots$ and $1.6787\ldots$, respectively. The proof obtains these bounds by expressing $a_2,\ldots,a_5$ in terms of the Grunsky coefficients of the odd function $\sqrt{f(z^2)}$, applying Grunsky’s inequality to control those coefficients, and maximizing the resulting functions $F_1$ and $F_2$ over the domain $D_1$. The reported bounds improve Theorem 1 of [7], which only gave $3.666\ldots$ and $3.2588\ldots$.

Load-bearing premise

The load-bearing assumption is that numerically solving the systems $\partial F_1/\partial x=\partial F_1/\partial y=0$ and $\partial F_2/\partial x=\partial F_2/\partial y=0$ really uncovers all interior maxima of $F_1$ and $F_2$ on $D_1$, and that the boundary checks are complete; the paper states this can be verified numerically but gives no proof or code.

Editorial extensions

If this is right

Every univalent function has $|a_2 a_4 - a_3^2| \leq 1.3614\ldots$.
Every univalent function has $|H_3(1)| \leq 1.6787\ldots$.
These values replace the previous best bounds $3.666\ldots$ and $3.2588\ldots$, more than halving the old estimates.
The reduction to the explicit functions $F_1$ and $F_2$ turns each Hankel estimate into a two-variable maximization problem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The numerical verification of the two critical-point systems could be upgraded to a rigorous computer-assisted proof using interval arithmetic, removing the one non-analytic step.
The same Grunsky-coefficient reduction may apply to other subclasses of univalent functions or to higher-order Hankel determinants, where the coefficient count grows but the structure persists.
The constants $1.3614$ and $1.6787$ are probably not sharp; the method does not identify extremal functions, and sharper bounds may exist.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First, the good news: the paper delivers what it promises. The bounds for |H2(2)| and |H3(1)| over the full class S are improved substantially (3.666 to 1.3614, and 3.2588 to 1.6787), and the proof is a clean Grunsky-coefficient argument. The reduction to the two-variable functions F1 and F2 is transparent, the use of the Grunsky inequalities is legitimate, and there is no circularity. I checked the algebra in Section 2; apart from a small coefficient slip discussed below, it works.

The soft spot is the global maximization of F1 and F2. For F1, the interior critical-point claim is actually easy to verify analytically, so that part is fine. For F2, the paper says the system has only two real interior solutions, but gives no code, no interval-arithmetic certificate, and no algebraic elimination. The boundary estimates are asserted without proof, though those are simple to check. The interior count is load-bearing: if there is an unobserved interior critical point with F2 above 1.6787, the claimed |H3(1)| bound does not follow. Spot evaluations near the reported point are consistent with the claimed maximum, so I suspect the result is true, but the proof as written is conditional on an unverified computation.

Minor point: in the reduction to F2, the terms 4xy and 2x^3 in the square-root factor should carry a 1/√5 from the |ω15| estimate. The stated F2 is coarser but still a valid upper bound, so it does not hurt the result; it just wastes a little slack.

Bottom line: this is a real improvement by a standard method, with one addressable gap. It deserves a serious referee who can either demand a rigorous certification of the critical-point count or accept a reproducible computational certificate. I would not cite the H3 bound as a proven theorem until that gap is closed, but the paper should definitely go to review.

Referee Report

3 major / 5 minor

Summary. The paper studies the Hankel determinants H2(2) and H3(1) for the class S of normalized univalent functions. Using the Grunsky coefficient representation (7) and the Grunsky inequality (3), the authors express H2(2) and H3(1) in terms of four Grunsky coefficients, apply the modulus bounds (9) and the estimate |a3-a2^2|<=1, and reduce the problem to maximizing the explicit functions F1 and F2 on the domain D1. They report that the interior critical systems have one (respectively two) real solutions and find the boundary maxima, leading to Theorem 2: |H2(2)| <= 1.3614... and |H3(1)| <= 1.6787..., improving the earlier bounds 11/3 and 3.2588... from [7].

Significance. Assuming the numerical maximization claims can be made rigorous, the result is a genuine and welcome improvement for the class S, where sharp estimates are not known. The algebraic reduction from Grunsky relations to F1 and F2 is mostly correct, and the use of (9) and |a3-a2^2|<=1 is standard. The paper is not circular: the bounds are consequences of the Grunsky inequalities, not fitted to the target constants. However, the proof as written is conditional on unverified numerical statements about global maxima; without a rigorous certification the theorem is not established. The paper also contains no code or machine-checkable artifacts, so the numerical claims cannot be independently reproduced from the text.

major comments (3)

[§2, after (17)] The assertion that the system dF2/dx=0, dF2/dy=0 has only two real solutions in the interior of D1 is the load-bearing step for the bound |H3(1)| <= 1.6787. The text supports this only by 'Numerical calculation give...' and reports the values at the two points. A missed interior critical point with F2 larger than the reported 1.6787... would invalidate Theorem 2(ii). Please replace this with a rigorous argument: for example, reduce the two equations to a univariate polynomial and use Sturm sequences or resultant computations, or provide an interval-arithmetic certified computation with the code or scripts used, or give an analytic proof. The boundary analysis alone does not remove this gap.
[§2, after (11)] The analogous statement in part (i), that dF1/dx=0, dF1/dy=0 has only one real solution in the interior of D1, is also made on the basis of numerical verification. This case is much easier: the equations reduce analytically to x^2=11/30 and y^2=281/1800, so the uniqueness and the value F1=1.19889... can be proven in a few lines. Since this point is part of the justification for the H2 bound, the derivation should be included rather than left to numerical assertion.
[§2, boundary of D1 in part (ii)] The boundary-maximum arguments for F2 also contain unproved monotonicity claims, in particular dF2/dx(x,0)<0 for 0<x<1 and dF2/dy(0,y)<=0 for 0<=y<=1/sqrt(3). These are elementary but load-bearing: they are how the text concludes F2(x,0)<=F2(0,0) and F2(0,y)<=F2(0,0). Please supply the derivative calculations or a certified interval check. This is a smaller gap than the interior critical-point count, but it should be closed as well.

minor comments (5)

[§1, equations (8)-(9)] In the derivation of (8)-(9), the inequality '5|omega15|^2+7|omega15|^2 <= 1' should read '5|omega15|^2+7|omega17|^2 <= 1'.
[Theorem 2] Theorem 2 labels both parts as '(i)'; the second should be '(ii)'.
[Throughout] There are several typos: 'NExt' before (10), 'satrt' after (15), 'seams' in Section 1, and 'univalent function' in reference [1]. These should be corrected.
[§2, reduction to F2 after (16)] In the reduction to F2, the terms (4xy+2x^3)*sqrt(...) are obtained by bounding |omega15| by sqrt(1-x^2-3y^2), which is weaker than the estimate in (9) and drops the factor 1/sqrt(5). The resulting F2 is a valid upper bound, but the weaker estimate should be stated explicitly to avoid appearing as a factor error.
[§2, boundary analysis, part (ii), item 3] The maximum of 2/(3*sqrt(3))*(1-x^2)^{3/2}+x^2(1-x^2) on [0,1] is quoted as 7/16 at x=1/2 without derivation; a one-line derivative check would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Hankel bounds are obtained from external Grunsky inequalities, not from the target values.

full rationale

The derivation chain is self-contained in the relevant sense: the bounds in Theorem 2 are obtained by bounding |H2(2)| and |H3(1)| in terms of Grunsky coefficient moduli via the external inequalities (3), (6), and (9), then maximizing explicit functions F1 and F2 on D1. The target constants 1.3614... and 1.6787... are outputs of those maximizations, not inputs; no parameter is fitted to the claimed bounds and no target value appears in the assumptions. The only caveats are the unverified numerical assertions that the critical-point systems for F1 and F2 have only one/two real solutions in the interior of D1 (Section 2, after (12) and (17)); this is a rigor gap concerning global maximization, not a circularity, because the same inequalities would remain valid if a different maximizer were found. Self-citations [4]-[8] are contextual (definition of U, prior bounds, conjectures) and none is load-bearing for the proof.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim relies on standard Grunsky theory, a reference relation with a claimed typo fix, and an unproven numerical maximization. There are no fitted parameters.

assumptions (4)

standard math Grunsky inequality (3): sum_q q |sum_p omega_p,q x_p|^2 <= sum_p |x_p|^2/p for arbitrary x_p
Invoked in Section 1 to derive the bounds (9) on Grunsky coefficients.
domain assumption Coefficient relations (7) from Lebedev [3, p.57] expressing a2,...,a5 in terms of Grunsky coefficients, including the authors' correction of a typo in a5 (5 omega_13^2 instead of 5 omega_15^2)
Used throughout Section 2 to compute H2(2) and H3(1). The correction is claimed but not independently verified here.
standard math Fekete-Szego inequality |a3 - a2^2| <= 1 for f in S
Used in part (ii) to bound |2 omega_13 - omega_11^2|.
ad hoc to paper Numerical verification that the systems of equations for interior critical points of F1 and F2 have only the stated solutions in D1
This is asserted without proof or code in Section 2 and is load-bearing for both parts of Theorem 2.

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Pith. "Pith review of An improvement of the estimates of the modulus of the Hankel determinants of second and third order for the class $\mathcal{S}$ of univalent functions." pith.science (2026). https://pith.science/paper/L5XKIS4Y

@misc{pith2026241112378,
  author       = {Pith},
  title        = {Pith review of: An improvement of the estimates of the modulus of the Hankel determinants of second and third order for the class $\mathcalS$ of univalent functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L5XKIS4Y}},
  note         = {Machine review of arXiv:2411.12378}
}

abstract

Using some properties of the Grunsky coefficients we improve earlier results for upper bounds of the Hankel determinants of the second and third order for the class $\mathcal{S}$ of univalent functions.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Symmetric Toeplitz determinants of some classes of univalent functions
math.CV 2025-09 reject novelty 4.0 of 10

Estimates for Toeplitz determinants of univalent functions are derived, but the claimed sharp bound 3/16 for T3,2 in class U with a2=0 is false; the true value is 1/4.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · cited by 1 Pith paper

[7]

Obradovi´ c and N

M. Obradovi´ c and N. Tuneski,Hankel determinants of second and third order for the class S of univalent functions, Math. Slovaca 71(3), 649-654, 2021

work page 2021
[1]

Duren, Univalent function, Springer-Verlag, New Yo rk, 1983

P.L. Duren, Univalent function, Springer-Verlag, New Yo rk, 1983

work page 1983
[2]

Janteng, S.A

A. Janteng, S.A. Halim and M. Darus, Hankel determinant fo r starlike and convex functions, Int. J. Math. Analysis, Vol. 1, No. 13, 2007, 619–225

work page 2007
[3]

Lebedev, Area principle in the theory of univalent fu nctions, Published by”Nauka”, Moscow, 1975 (in Russian)

N.A. Lebedev, Area principle in the theory of univalent fu nctions, Published by”Nauka”, Moscow, 1975 (in Russian)

work page 1975
[4]

Obradovi´ c and S

M. Obradovi´ c and S. Ponnusamy, New criteria and distortion theorems for univalent func- tions, Complex Variables Theory Appl. 44(2001), 173–191. (Also Reports of the Department of Mathematics, Preprint 190, June 1998, University of Hels inki, Finland)

work page 2001
[5]

Obradovi´ c and S

M. Obradovi´ c and S. Ponnusamy, On the class U, Proc. 21st Annual Conference of the Jammu Math. Soc. and a National Seminar on Analysis and its Ap plication, 11–26, 2011

work page 2011
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Obradovi´ c and N

M. Obradovi´ c and N. Tuneski, Some properties of the class U, Ann. Univ. Mariae Curie- Sk/suppress lodowska, 73(1), 49-56, 2019

work page 2019
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New upper bounds of the third Hankel determinant for some classes of univalent functions

M. Obradovi´ c and N. Tuneski, New upper bounds of the thirdHankel determinant for some classes of univalent functions, submitted. (https://arxi v.org/abs/1911.10770)

work page Pith review arXiv 1911

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[9]

Dienes P., The Taylor Series: An Introduction to the Theor y of Functions of a Complex Variable; NewYork-Dover: Mineola, NY, USA, 1957

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[10]

Shi L., Srivastava H.M., Arif M., Hussain S., Khan H., An I nvestigation of the Third Hankel Determinant Problem for Certain Subfamilies of Univalent F unctions Involving the Expo- nential Function, Symmetry 2019, 11, 598. Department of Mathematics, F aculty of Civil Engineeri...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [7]

Obradovi´ c and N

M. Obradovi´ c and N. Tuneski,Hankel determinants of second and third order for the class S of univalent functions, Math. Slovaca 71(3), 649-654, 2021

work page 2021

[2] [1]

Duren, Univalent function, Springer-Verlag, New Yo rk, 1983

P.L. Duren, Univalent function, Springer-Verlag, New Yo rk, 1983

work page 1983

[3] [2]

Janteng, S.A

A. Janteng, S.A. Halim and M. Darus, Hankel determinant fo r starlike and convex functions, Int. J. Math. Analysis, Vol. 1, No. 13, 2007, 619–225

work page 2007

[4] [3]

Lebedev, Area principle in the theory of univalent fu nctions, Published by”Nauka”, Moscow, 1975 (in Russian)

N.A. Lebedev, Area principle in the theory of univalent fu nctions, Published by”Nauka”, Moscow, 1975 (in Russian)

work page 1975

[5] [4]

Obradovi´ c and S

M. Obradovi´ c and S. Ponnusamy, New criteria and distortion theorems for univalent func- tions, Complex Variables Theory Appl. 44(2001), 173–191. (Also Reports of the Department of Mathematics, Preprint 190, June 1998, University of Hels inki, Finland)

work page 2001

[6] [5]

Obradovi´ c and S

M. Obradovi´ c and S. Ponnusamy, On the class U, Proc. 21st Annual Conference of the Jammu Math. Soc. and a National Seminar on Analysis and its Ap plication, 11–26, 2011

work page 2011

[7] [6]

Obradovi´ c and N

M. Obradovi´ c and N. Tuneski, Some properties of the class U, Ann. Univ. Mariae Curie- Sk/suppress lodowska, 73(1), 49-56, 2019

work page 2019

[8] [8]

New upper bounds of the third Hankel determinant for some classes of univalent functions

M. Obradovi´ c and N. Tuneski, New upper bounds of the thirdHankel determinant for some classes of univalent functions, submitted. (https://arxi v.org/abs/1911.10770)

work page Pith review arXiv 1911