Improvements of certain results of the class $\mathcal{S}$ of univalent functions

Milutin Obradovi\'c; Nikola Tuneski

REVIEW 2 major objections 4 minor 1 cited by

Improvements of certain results of the class $\mathcal{S}$ of univalent functions

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Pith's one-line read This paper improves the known upper bounds on the second and third Hankel determinants of univalent functions when either the quadratic or cubic coefficient vanishes, and also improves the bound on |a4|-|a3|.

desk verdict A useful but flawed manuscript: the a2=0 bounds and the |a4|−|a3| improvement look right, but the advertised a3=0 third Hankel bound rests on an algebra error. read the letter →

arxiv 2411.13102 v1 pith:W5LK5KRL submitted 2024-11-20 math.CV

Milutin Obradović , Nikola Tuneski This is my paper · ORCID

classification math.CV MSC 30C4530C5030C55

keywords univalentfunctionsGrunskycoefficientsHankeldeterminantsecondthirdcoefficientdifferenceestimates

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 studies the class $\mathcal{S}$ of functions that are univalent (one-to-one) in the unit disk and expand as $f(z)=z+a_2z^2+a_3z^3+\cdots$. The authors improve the known upper bounds for the second Hankel determinant $H_2(2)=a_2a_4-a_3^2$ and the third Hankel determinant $H_3(1)=a_3(a_2a_4-a_3^2)-a_4(a_4-a_2a_3)+a_5(a_3-a_2^2)$ in the two special cases where either $a_2=0$ or $a_3=0$. They also improve the bound on the coefficient difference $|a_4|-|a_3|$ for the whole class $\mathcal{S}$, and give a new estimate for $|a_5|-|a_3|$ for odd functions. The bounds are not claimed to be sharp, but each narrows a long-open gap in the coefficient theory of univalent functions.

What carries the argument

The central object is the set of Grunsky coefficients $\omega_{p,q}$, defined by $\log\frac{f(t)-f(z)}{t-z}=\sum_{p,q\ge0}\omega_{p,q}t^p z^q$. For the odd square-root transform $f_2(z)=\sqrt{f(z^2)}$, identities (6) express $a_2,\dots,a_5$ in terms of $\omega_{11},\omega_{13},\omega_{33},\omega_{15},\omega_{35},\omega_{17}$, and the Grunsky inequality (4), truncated to (7), yields the size constraints (8) on the $\omega$'s. Substituting the identities into $H_2(2)$, $H_3(1)$, and the coefficient differences reduces each desired bound to the maximum of an explicit single- or two-variable function $F_1,\dots,F_6$ on a compact domain; the maxima are then located by the first derivative test.

What would settle it

Directly expand $H_3(1)=-a_4^2-a_5a_2^2$ using the paper's own formulas $a_2=2\omega_{11}$, $a_4=2\omega_{15}-5\omega_{11}^3$, $a_5=2\omega_{17}+6\omega_{11}\omega_{15}-\frac{25}{4}\omega_{11}^4$. The coefficient of $\omega_{15}^2$ is $-4$, not $-1$, giving $-4\omega_{15}^2-4\omega_{11}^3\omega_{15}-8\omega_{11}^2\omega_{17}$; maximizing $|\cdot|$ over $|\omega_{11}|\le 1/2$ with the constraints (18) and (21) gives a number to compare with $0.6647958756\ldots$.

Watch

Extended reading notes

Core claim

The paper proves several new estimates. For $f\in\mathcal{S}$ with $a_2=0$, it shows $|a_5|\le \frac34+\frac1{\sqrt7}=1.12796\ldots$ and $|H_3(1)|\le 1.026\ldots$, improving the earlier $|a_5|\le 1.508\ldots$ and $|H_3(1)|\le 2.05$. For $f\in\mathcal{S}$ with $a_3=0$, it shows $|a_2|\le 1$, $|a_4|\le \frac14\sqrt{\frac{21}{5}}+\frac58=1.1373\ldots$, $|a_5|\le 1.674896577\ldots$, $|H_2(2)|\le 1.1373\ldots$, and $|H_3(1)|\le 0.6647958756\ldots$, the last improving the previous bound $1.114596\ldots$. For general $f\in\mathcal{S}$, it proves $|a_4|-|a_3|\le 1.75185\ldots$, improving the previous $2.1033299\ldots$, and for odd functions $|a_5|-|a_3|\le 2/\sqrt7=0.7559\ldots$. All of these constants come from a single framework: translate the coefficients $a_2,\ldots,a_5$ into Grunsky coefficients and maximize the resulting explicit functions.

Load-bearing premise

The whole argument hinges on the Grunsky-coefficient identities (6) and on the algebraic expansions derived from them, especially the expression (23) that converts $H_3(1)$ into a polynomial in $\omega_{11},\omega_{15},\omega_{17}$; if those identities or the expansions are not correct, the improved bounds for the third Hankel determinant do not follow.

Editorial extensions

If this is right

For $a_2=0$, the third Hankel determinant bound drops from $2.05$ to $1.026\ldots$, and the fifth-coefficient bound from $1.508\ldots$ to $1.12796\ldots$.
For $a_3=0$, $|H_3(1)|\le 0.6647958756\ldots$ replaces $1.114596\ldots$, and $|H_2(2)|\le 1.1373\ldots$ replaces $1.75088\ldots$.
The coefficient-difference bound $|a_4|-|a_3|\le 1.75185\ldots$ improves the best known $2.1033299\ldots$ for $n=3$.
Odd functions satisfy $|a_5|-|a_3|\le 2/\sqrt7=0.7559\ldots$, an improvement over the earlier $|a_5|-|a_3|<1$.
The corrected formula for $a_5$ in the Grunsky expansion removes a typo from the literature, so any result built on the old expression should be re-examined.

Reading between the lines

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

The same reduction to a finite-dimensional optimization should extend to higher Hankel determinants $H_4(1)$ or $H_5(1)$, but the number of Grunsky coefficients and the algebraic complexity grow quickly; the $a_2=0$ and $a_3=0$ cases are the natural first test.
Because the derivative tests place the maxima of $F_2,F_3,F_4,F_5$ inside the intervals, the extremal functions are likely not Koebe-type maps; identifying them exactly would settle the sharpness of the new bounds.
The corrected $a_5$ identity may propagate: other papers that quoted the erroneous term $5\omega_{15}^2$ instead of $5\omega_{13}^2$ will need their estimates recalculated.
Using the full Grunsky inequality rather than the truncated form (7) could in principle sharpen the constraints (8) further, turning the finite optimization into an infinite-dimensional one.

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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

Obradović and Tuneski improve several coefficient bounds in S using Grunsky coefficients. The a2=0 case (Theorem 4) gives |a5| ≤ 0.75 + 1/√7 and |H3(1)| ≤ 1.026, and the proof is clean. Theorem 6's improvement of |a4|−|a3| to 1.75185 also checks out as a genuine step forward. The a3=0 parts (i)–(iv) look correct.

The problem is Theorem 5(v). Equation (23) is algebraically inconsistent with the paper's own formulas. Substituting a2=2ω11, a4=2ω15−5ω11^3, a5=2ω17+6ω11ω15−(25/4)ω11^4 into H3(1) = −a4^2 − a5a2^2 gives −4ω15^2 −4ω11^3ω15 −8ω11^2ω17, not −ω15^2 −4ω11^3ω15 −8ω11^2ω17. The printed bound F5 uses the wrong coefficient (1/5 instead of 4/5), so the maximum 0.6647958756 is not proved. Re-running the triangle estimates with the corrected coefficient yields a bound around 1.02, which still beats the old 1.114596 from [9] but does not support the advertised number. The abstract's claim of improved third Hankel bounds in the a3=0 case is therefore not established as written.

This is a localized but load-bearing error. The rest of the derivations are standard chains of inequalities, and the numerical maximizations are reported without derivative checks, but that is typical in this area and the algebra is easy to verify. The paper belongs in a revision rather than a desk reject: the authors are clearly serious and the corrected bound, if reworked, would still be an improvement. I would send it to peer review with a note to check the algebra in Theorem 5(v) and recompute F5. Specialists in Hankel determinants for S will want to read the a2=0 and coefficient-difference parts, but should not trust Theorem 5(v)'s headline number until it is fixed.

Referee Report

2 major / 4 minor

Summary. The paper derives new upper bounds for |a5| and |H3(1)| for functions in the class S of univalent functions on the unit disk, in the two special cases a2=0 and a3=0, together with an improved estimate for |a4|-|a3|. The proofs use the Grunsky coefficient representation of the initial coefficients and Grunsky's inequality, reducing the bounds to one- or two-variable extremal problems. Theorems 4 and 5 present the improved Hankel determinant bounds; Theorem 6 gives the coefficient-difference bound and an odd-function result.

Significance. If valid, the results are modest but genuine improvements over the previous bounds from [9], and the paper correctly identifies a misprint in [9] for |a5| when a2=0. The method is standard, and most of the derivations are straightforward. However, one of the advertised improvements, the bound for H3(1) in the a3=0 case, is invalid as written because of an algebraic error, so the paper's central claim needs substantial revision.

major comments (2)

[Theorem 5(v), Eq. (23)] The expansion of H3(1) in Eq. (23) is algebraically incorrect. Substituting a2=2ω11, a4=2ω15-5ω11^3, and a5=2ω17+6ω11ω15-(25/4)ω11^4 (as derived in Eqs. (17) and (20)) into Eq. (22) gives H3(1)=-(2ω15-5ω11^3)^2-4ω11^2(2ω17+6ω11ω15-(25/4)ω11^4)=-4ω15^2-4ω11^3ω15-8ω11^2ω17, not the displayed -ω15^2-4ω11^3ω15-8ω11^2ω17. Consequently the function F5 and the claimed maximum 0.6647958756 are not consequences of the preceding equations. This invalidates the proof of Theorem 5(v), one of the advertised improvements in the abstract.
[Theorems 5(iii), 5(v), 6(i)] The numerical maximizations are not rigorously justified. For the one-variable functions in Theorems 5(iii) and 5(v), the text merely states that "the first derivative test shows" the maximum, without presenting the derivative or proving that the stated critical point is the unique maximizer on the interval. For the two-variable function F6 in Theorem 6(i), the paper numerically solves a system to find a stationary point but does not prove that this exhausts all stationary points in D1, so the claim that F6 attains its maximum there is not established. Please provide complete derivative sign analyses or other rigorous arguments.

minor comments (4)

[Theorem 6(i), proof] In the displayed inequality chain, the term "≤ |ω15|+4|ω11||ω13|+|ω11|^3" should read "≤ 2|ω15|+4|ω11||ω13|+|ω11|^3", and the following line should end with "|ω11|^3" rather than "|ω13|^3", to match the definition of F6(x,y).
[References] Reference [8] appears incomplete: "An improvement fo the Hankel determinants..." lacks publication details.
[Before Theorem 6] The stray line "*************************** DTS" appearing before Theorem 6 should be removed.
[Eq. (14)] The notation "ω11^2 = -2/3 ω13" is an equality of complex numbers; the biconditional "⇔" is misleading and should be rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the paper's derivations are self-contained via external Grunsky inequalities.

full rationale

The paper's claimed improvements are derived from Grunsky coefficient identities (6) and inequalities (8) imported from the external reference [5], together with the classical |a3-a2^2|<=1 inequality. None of the target bounds is assumed or fitted. Each theorem in the main results section constructs an explicit majorizing function (e.g., F1 through F6) and maximizes it over a domain obtained from Grunsky inequalities; the resulting values are consequences of the Grunsky inequalities, not self-referential. The paper cites its own previous results [9] as baselines to compare with, but does not rely on them as premises for the proofs. There is, however, an apparent algebraic error in the derivation of Eq. (23) in Theorem 5(v): substituting the paper's own a4=2ω15-5ω11^3 and a5=2ω17+6ω11ω15-(25/4)ω11^4 into (22) yields H3(1)=-4ω15^2-4ω11^3ω15-8ω11^2ω17, not -ω15^2-4ω11^3ω15-8ω11^2ω17. That affects the correctness of the numerical bound 0.6647958756, but this is a computational/typing error rather than a circular dependency of the kind assessed here. No circular step satisfies the requirement of being reducible by the paper's own equations to its inputs.

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

The paper introduces no free parameters or invented entities. Its entire burden rests on standard Grunsky coefficient theory and on the algebraic correctness of the coefficient-to-Grunsky identities; the algebraic correctness fails at Eq (23).

assumptions (4)

standard math Grunsky inequality (4) holds for all f in S.
Imported from Duren and Lebedev; the foundation of every estimate in the paper.
standard math Coefficient formulas (6) express a2 through a5 via Grunsky coefficients of f2.
Taken from Lebedev [5] with a correction to a5; used to convert coefficient functionals into Grunsky coefficient expressions.
standard math Inequalities (8) follow from (7) by setting x1=1 and x3=0.
Used throughout the paper to bound |ω13|, |ω15|, and |ω17| in terms of |ω11|.
standard math Fekete-Szegő inequality |a3 - a2^2| ≤ 1.
Used in Theorem 4(i) and Theorem 5(i).

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Pith. "Pith review of Improvements of certain results of the class $\mathcal{S}$ of univalent functions." pith.science (2026). https://pith.science/paper/W5LK5KRL

@misc{pith2026241113102,
  author       = {Pith},
  title        = {Pith review of: Improvements of certain results of the class $\mathcalS$ of univalent functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W5LK5KRL}},
  note         = {Machine review of arXiv:2411.13102}
}

abstract

For $f\in \mathcal{S}$, the class univalent functions in the unit disk $\mathbb{D}$ and given by $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$ for $z\in \mathbb{D}$, we improve previous bounds for the second and third Hankel determinants in case when either $a_2=0,$ or $a_3=0$. We also improve an upper bound for the coefficient difference $|a_4|-|a_3|$ when $f\in \mathcal{S}$.

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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

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

[9]

Milutin Obradovi\' c and Nikola Tuneski, An improvement fo the Hankel determinants of second and third order for the class S of univalent functions ,

work page
[1]

De Branges, A proof of the Bieberbach conjecture, Acta Math

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[2]

Duren, Univalent function, Springer-Verlag, New York, 1983

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

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[3]

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[4]

A. Z. Grinspan, The sharpening of the difference of the moduli of adjacent coefficients of schlicht functions, In Some problems in modern function theory Proc. Conf. Modern Problems of Geometric Theory of Functions, Inst. Math., Acad. Sci. USSR, Novosibirsk, (Russian), Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk, (1976), 41--45

work page 1976
[5]

W. K. Hayman, On successive coefficients of univalent functions, J. London Math. Soc., 38 (1963), 228--243

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[6]

Lebedev, Area principle in the theory of univalent functions, Published by"Nauka", Moscow, 1975 (in Russian)

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

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[7]

Lecko, D

A. Lecko, D. Partyka, A generalized Fekete-Szegö functional and initial successive coefficients of univalent functions, Bulletin des Sciences Mathématiques, 197 (2024), art. id. 103527

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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

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D.K. Thomas, N. Tuneski, A. Vasudevarao, Univalent Functions: A Primer, De Gruyter Studies in Mathematics 69 , De Gruyter, Berlin, Boston, 2018

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Pith tools

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

[1] [9]

Milutin Obradovi\' c and Nikola Tuneski, An improvement fo the Hankel determinants of second and third order for the class S of univalent functions ,

work page

[2] [1]

De Branges, A proof of the Bieberbach conjecture, Acta Math

L. De Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), no. 1--2, 137--152

work page 1985

[3] [2]

Duren, Univalent function, Springer-Verlag, New York, 1983

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

1983

[4] [3]

o, Eine Bemerkung \

M. Fekete and G. Szeg\"o, Eine Bemerkung \"uber ungerade schlichte Funktionen, J. London Math. Soc. 8 (1933), 85--89

work page 1933

[5] [4]

A. Z. Grinspan, The sharpening of the difference of the moduli of adjacent coefficients of schlicht functions, In Some problems in modern function theory Proc. Conf. Modern Problems of Geometric Theory of Functions, Inst. Math., Acad. Sci. USSR, Novosibirsk, (Russian), Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk, (1976), 41--45

work page 1976

[6] [5]

W. K. Hayman, On successive coefficients of univalent functions, J. London Math. Soc., 38 (1963), 228--243

work page 1963

[7] [6]

Lebedev, Area principle in the theory of univalent functions, Published by"Nauka", Moscow, 1975 (in Russian)

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

work page 1975

[8] [7]

Lecko, D

A. Lecko, D. Partyka, A generalized Fekete-Szegö functional and initial successive coefficients of univalent functions, Bulletin des Sciences Mathématiques, 197 (2024), art. id. 103527

work page 2024