Symmetric Toeplitz determinants of some classes of univalent functions

Milutin Obradovi\'c; Nikola Tuneski

REVIEW 3 major objections 4 minor 15 references

Symmetric Toeplitz determinants of some classes of univalent functions

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper establishes sharp upper bounds for symmetric Toeplitz determinants of univalent functions in the class U, with explicit extremal functions.

desk verdict A routine coefficient-determinant paper with one false sharp constant; the rest is mostly elementary and correctable. read the letter →

arxiv 2509.09385 v1 pith:RPXUSNIV submitted 2025-09-11 math.CV

Milutin Obradović , Nikola Tuneski This is my paper · ORCID

classification math.CV MSC 30C4530C5030C55

keywords ToeplitzdeterminantunivalentfunctionsclassUsharpboundscoefficientestimatesSchwarzHankel

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 symmetric Toeplitz determinants built from the Taylor coefficients of univalent functions, specifically the class U defined by a differential inequality. It establishes upper bounds for several determinants of orders 2 and 3, and shows that most of these bounds are best possible by constructing extremal functions. The sharpest claimed result is for T_{3,2} when the second coefficient vanishes: the paper asserts a bound of 3/16. The proofs rest on a parametrization of U by Schwarz functions, which yields explicit coefficient estimates. If the bounds hold, they give the exact largest possible modulus of these determinants within the class.

What carries the argument

The central object is the symmetric Toeplitz determinant T_{q,n}(f), defined as the determinant of a Toeplitz matrix with entries a_n,...,a_{n+q-1}. The proofs use Lemma 1, which parametrizes f in U as z/f = 1 − a2 z − z ω(z) with ω a Schwarz function, giving the coefficient identities a3=a2^2+c1, a4=c2+2 a2 c1+a2^3, a5=c3+2 a2 c2+c1^2+3 a2^2 c1+a2^4, together with the bounds |c1|≤1, |c2|≤(1−|c1|^2)/2, and a bound on c3. Combining these identities with known Hankel determinant estimates and the factorization formulas (3) produces each upper bound.

What would settle it

Evaluate the paper's extremal function f4 defined by z/f4 = 1 − z times the integral from 0 to z of (1/sqrt(2)+t)/(1+t/sqrt(2)) dt. It has a2=0 and |c1|=1/sqrt(2), and substituting into |T_{3,2}|=2|a3|^2|a4| yields 2·(1/2)·(1/4)=1/4. Since 1/4 > 3/16, the claimed sharp bound is falsified.

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Extended reading notes

Core claim

For a function f(z)=z+a2 z^2+... in the class U, the paper proves that |T_{2,2}(f)|≤13, |T_{2,3}(f)|≤25, |T_{3,1}(f)|≤24, |T_{3,2}(f)|≤84, and |T_{3,3}(f)|≤211.88, with the first four sharp. For functions in U with a2=0, it proves |T_{2,2}|≤1, |T_{2,3}|≤1, |T_{3,1}|≤2, |T_{3,2}|≤3/16, and |T_{3,3}|≤9/2, again with the first four sharp; the extremal functions include z/(1−z^2) and an integral-defined function z/f4 = 1 − z times the integral from 0 to z of (1/sqrt(2)+t)/(1+t/sqrt(2)) dt. For the general univalent class S it gives |T_{3,2}|≤86.17 and |T_{3,3}|≤239.19, and with a2=0 the bounds 4/3 and 7.39. The determinant T_{q,n} is a Toeplitz matrix whose entries are consecutive Taylor coeffic

Load-bearing premise

The proof of Theorem 2(iv) assumes that |c1|^2(1−|c1|^2)≤3/16 for |c1|≤1, but that product's true maximum is 1/4.

Editorial extensions

If this is right

The bounds in Theorem 1, if valid, are the exact maxima of |T_{q,n}| over U for (q,n) = (2,2), (2,3), (3,1), and (3,2).
For the a2=0 subclass, the paper asserts even smaller sharp bounds, including 3/16 for T_{3,2}.
Explicit extremal functions are given: f1(z)=z/(1−iz)^2 for the generic U bounds and f2(z)=z/(1−z^2), f3(z)=z/(1−iz^2), and an integral-defined f4 for the a2=0 sharp cases.
The estimates for the general class S are derived by combining the same factorization with known Hankel determinant bounds.
These Toeplitz determinant bounds complement the existing sharp Hankel determinant results for the same classes.

Reading between the lines

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

Applying the paper's own coefficient bound |c2| ≤ (1−|c1|^2)/2 to the equality |T_{3,2}| = 2|a3|^2|a4| = 2|c1|^2|c2| yields a maximum of t(1−t) ≤ 1/4 on t=|c1|^2∈[0,1]. The claimed sharp value 3/16 is therefore not supported by the proof, and the extremal f4 attains 1/4.
The same parametrization scheme could be applied to higher-order Toeplitz determinants T_{q,n} with q,n≥4, once analogous factorization formulas are derived.
A natural testable extension is to consider the class defined by |(z/f)^2 f' − 1| < λ for general λ>0 and compute how the sharp Toeplitz bounds scale with λ.

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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

Colleague, quick take on arXiv:2509.09385. The reader's rejection is right. The central problem is Theorem 2(iv): for f in U with a2=0, the claimed |T3,2| <= 3/16 is wrong. From the paper's own Lemma 1, |c2| <= (1-|c1|^2)/2, so T3,2 = 2|c1|^2|c2| <= |c1|^2(1-|c1|^2), whose maximum is 1/4. The extremal f4 they write down attains 1/4, not 3/16. So the sharp bound they advertise is false, and the stated inequality t(1-t) <= 3/16 is simply wrong. That's not a typo; it's load-bearing.

What's actually new: the paper extends the Toeplitz determinant program to q=3, n=2 and q=3, n=3 for class U and for the full class S, and gives a2=0 variants. The derivations are one-line substitutions of known coefficient and Hankel bounds from their earlier papers, so the novelty is modest but legitimate. The sharpness examples for the other bounds look right, and the general technique is sound. The paper is honest about its inputs: Lemmas 2 and 3 come from earlier work, so any correction in those inputs will propagate.

The other soft spots are smaller but real. Theorem 1(v) uses 1.4846575 in the computation while Lemma 2(b) says 1.4946575, and the printed constant 211.8771 matches neither 211.4846575 nor 211.95726. Theorems 3(ii) and 4(ii) are mislabeled: the proofs clearly give bounds for T3,3, not T2,3. These are fixable typos, but they shouldn't be in a finalized paper.

If Theorem 2(iv) is corrected to 1/4, the paper becomes a thin but valid addition to a niche literature. As it stands, the false sharp constant and the mislabeled theorems make it not acceptable as submitted. I'd send it back for major revision rather than desk-reject, because the remaining results are likely correct and the paper is short enough that a referee can verify the fix quickly. The audience is specialists in geometric function theory working on coefficient functionals; nobody else needs this. Reading group maybe, for a cautionary example of checking extremal functions. I'd not cite it until corrected.

Referee Report

3 major / 4 minor

Summary. The paper studies upper bounds for symmetric Toeplitz determinants T_{q,n}(f) for the class U of univalent functions satisfying |(z/f)^2 f' - 1| < 1 and for the full class S. For U it states bounds for T_{2,2}, T_{2,3}, T_{3,1}, T_{3,2}, T_{3,3} (Theorem 1), and for the subclass with a_2=0 (Theorem 2). It also gives bounds for S (Theorems 3 and 4). The proofs combine coefficient estimates from Lemma 1 with known Hankel determinant bounds from the authors' earlier work (Lemmas 2 and 3). Several results are claimed to be sharp, with explicit extremal functions.

Significance. If correct, the sharp estimates for the class U would be a modest contribution to the Toeplitz-determinant literature. The paper is short and computational, but it does provide verifiable closed-form bounds and extremal examples (e.g., f_1 and f_2). However, the manuscript as written contains a false sharp constant, inconsistent numerical values, and mislabeled determinant indices in two theorems. These are not cosmetic issues: the main statements are wrong as printed. The method is standard, and the underlying approach can likely be repaired, but the current submission cannot be accepted without substantial correction.

major comments (3)

[Theorem 2(iv), proof] The stated sharp bound |T_{3,2}| ≤ 3/16 is false. With a_2=0, (3) gives T_{3,2} = 2a_3^2 a_4 = 2 c_1^2 c_2. Combining with |c_2| ≤ (1-|c_1|^2)/2 yields |T_{3,2}| ≤ |c_1|^2(1-|c_1|^2) = t(1-t), t∈[0,1]. The maximum is 1/4, not 3/16. The claimed extremal f_4 has c_1 = 1/√2 and c_2 = 1/4, giving |T_{3,2}(f_4)| = 2·(1/2)·(1/4) = 1/4. Thus the sharp constant should be 1/4, and the sentence 'equality is attained for |c_1|^2 = 1/2' supports 1/4, not 3/16.
[Theorem 1(v) and Lemma 2(b)] The numerical constant in Theorem 1(v) is inconsistent. Lemma 2(b) states |H_{2,3}(f)| ≤ 1.4946575..., but the proof of Theorem 1(v) uses 1.4846575... in the line |T_{3,3}(f)| ≤ 8·(25 + 1.4846575...) = 211.4846575... . With the lemma's value the bound becomes 8·(25+1.4946575) ≈ 211.9573, whereas the theorem states 211.8771. The source of the mismatch (possibly a typographical transposition) must be corrected, and the stated value aligned with the actual calculation.
[Theorems 3(ii) and 4(ii)] The statements claim bounds on |T_{2,3}(f)|, but the proofs bound |T_{3,3}(f)|. In Theorem 3(ii) the proof uses |T_{3,3}| ≤ (|a_3|+|a_5|)(|a_3|^2+|a_4|^2+|H_{2,3}|), which is the formula for T_{3,3}, not T_{2,3}. The same pattern appears in Theorem 4(ii). As written, the bounds 239.1895 and 7.3883 for T_{2,3} are impossible: since T_{2,3} = a_3^2 - a_4^2, the trivial bound for S is |a_3|^2+|a_4|^2 ≤ 25, and for a_2=0 it is ≤ 1+(2/3)^2 = 13/9. The index T_{3,3} should be used in both theorem statements, or the proofs and bounds must be replaced.

minor comments (4)

[Throughout] The notation for the extremal function in Theorem 2(iv) is inconsistent: the text uses both f4 and f_4. Use a single consistent subscripted form.
[Theorem 2(v) proof] The typo 'H2.3(f)' should be 'H_{2,3}(f)'. Also the bound |a_5| ≤ 1 is derived correctly, but the chain of inequalities in the proof of (v) is written in a compressed way; expanding it would improve readability.
[Proof of Theorem 1(iv)] The estimate |a_4| ≤ 4 for f∈U is used without citation. This is a standard coefficient bound for U, but for completeness a reference (or a one-line justification) should be given.
[Abstract] The abstract says 'for certain values of q and n' without being specific. Since the theorems cover precise q,n, listing them (e.g., (2,2),(2,3),(3,1),(3,2),(3,3)) would make the contribution clearer.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Toeplitz bounds are derived by substituting independent coefficient/Hankel estimates into algebraic identities; the false sharp constant in Theorem 2(iv) is a correctness error, not a circular one.

full rationale

The estimates in Theorems 1–4 are obtained by taking the algebraic formulas for T_{q,n}(f) in (3), substituting the coefficient relations (5) from Lemma 1, and then bounding the resulting terms with the coefficient and Hankel determinant estimates in Lemmas 2 and 3. No parameter is fitted to the target Toeplitz functional and no conclusion is assumed in its own proof. Lemma 2 and Lemma 3 are cited from the authors' own earlier papers ([8,10] and [10,13]), and Theorem 4 uses [9,14]; these citations are load-bearing for the numerical constants, but they concern Hankel determinants or coefficient bounds for U and S, not symmetric Toeplitz determinants, so under the stated assumptions they are independent inputs rather than a restatement of the target results. Self-citation alone is therefore not construction-level circularity. Separately, the proof of Theorem 2(iv) contains a genuine arithmetic defect: the chain "2|c1|^2|c2| ≤ 2|c1|^2 · 1/2 (1−|c1|^2) ≤ 3/16" uses max_{0≤t≤1} t(1−t)=3/16, but the actual maximum is 1/4, attained at t=1/2. The paper's own extremal f_4, with |c1|=1/√2 and c2=1/4, gives |T_{3,2}(f_4)| = 2·(1/2)·(1/4) = 1/4, not 3/16. This invalidates the stated sharp constant, but it is a correctness issue rather than a circularity issue, so it does not raise the circularity score. Overall: no significant circularity detected.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data and no new entities are introduced. The central claims rest entirely on prior coefficient and Hankel bounds, treated here as input axioms.

assumptions (5)

domain assumption Lemma 1: f in U iff z/f = 1 - a2 z - z omega(z) with |omega(z)|<1 and |omega'(z)|<=1, with the coefficient bounds (6).
Inherited from the authors' reference [8]; all class U estimates rest on this characterization.
domain assumption Coefficient bounds |a2|<=2, |a3|<=3, |a4|<=4 for functions in U.
Used without proof in Theorem 1 and taken from the known theory of the class U.
domain assumption Lemma 2 Hankel bounds for U: |H2,2|<=1, |H2,3|<=1.4946575 (or 1.4846575 in the proof), and |H2,3|<=1 when a2=0.
Self-cited prior results; Theorem 1(iv), Theorem 1(v), and Theorem 2(v) depend on them. The inconsistency between the two displayed constants is noted.
domain assumption Lemma 3 Hankel bounds for S: |H2,2|<=1.3614, |H2,3|<=4.89869, and |H2,3|<=2.02757 when a2=0.
Self-cited prior results; Theorems 3 and 4 depend on these bounds.
domain assumption Coefficient bounds for S with a2=0: |a3|<=1, |a4|<=2/3, |a5|<=1.12796.
Cited from references [9,14] and used in Theorem 4.

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Pith. "Pith review of Symmetric Toeplitz determinants of some classes of univalent functions." pith.science (2026). https://pith.science/paper/RPXUSNIV

@misc{pith2026250909385,
  author       = {Pith},
  title        = {Pith review of: Symmetric Toeplitz determinants of some classes of univalent functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RPXUSNIV}},
  note         = {Machine review of arXiv:2509.09385}
}

abstract

In this paper, we consider estimates of symmetric Toeplitz determinants $T_{q,n}(f)$ for the class ${\mathcal U}$ and for the general class ${\mathcal S}$ for certain values of $q$ and $n$ ($q,n=1,2,3\ldots$).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 3 linked inside Pith

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Thomas, A

Md Firoz Ali, D.K. Thomas, A. Vasudevarao,Toeplitz determinants whose elements are the coefficients of analytic and univalent functions, Bull. Aust. Math. Soc.97(2) (2018), 253–264

2018

[2] [2]

Hayman, On the second Hankel determinant of mean univalent functions,Proc

W.K. Hayman, On the second Hankel determinant of mean univalent functions,Proc. London Math. Soc.3(18) (1968), 77-–94

1968

[3] [3]

Janteng, S.A

A. Janteng, S.A. Halim, M. Darus, Coefficient inequality for a function whose derivative has a positive real part.J. Inequal. Pure Appl. Math.7(2) (2006), Article 50, 5 pp

2006

[4] [4]

Janteng, S.A

A. Janteng, S.A. Halim, M. Darus, Hankel determinant for starlike and convex functions,Int. J. Math. Anal. (Ruse).1(13-16) (2007), 619—625

2007

[5] [5]

Kowalczyk, A

B. Kowalczyk, A. Lecko, Y.J. Sim, The sharp bound of the Hankel determinant of the third kind for convex functions.Bull. Aust. Math. Soc.97(3) (2018), 435–445

2018

[6] [6]

Kowalczyk, A

B. Kowalczyk, A. Lecko, The second Hankel determinant of the logarithmic coefficients of strongly starlike and strongly convex functions,Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.,117, 91 (2023)

2023

[7] [7]

Obradovi´ c, S

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

2011

[8] [8]

Obradovi` c, N

M. Obradovi` c, N. Tuneski, Some properties of the classU, Ann. Univ. Mariae Curie- Sk lodowska Sect. A,73(1) (2019), 49–56

2019