Faber polynomial coefficient estimation of subclass of bi-subordinate univalent functions

Alaa H. El-Qadeem; Mohamed A. Mamon; S.A. Saleh

REVIEW 4 major objections 4 minor 23 references

Faber polynomial coefficient estimation of subclass of bi-subordinate univalent functions

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper derives explicit bounds on the second and third Maclaurin coefficients and the Fekete–Szegő functional for a generalized bi-univalent class via Faber polynomials.

desk verdict A legitimate new class and a correct Theorem 2.4, but the main Theorem 2.8 is broken by sign errors and impossible case conditions. read the letter →

arxiv 1908.07349 v6 pith:NF5OUR3O submitted 2019-08-08 math.CV

S.A. Saleh , Alaa H. El-Qadeem , Mohamed A. Mamon This is my paper · ORCID

classification math.CV MSC 30C4530C5030C55

keywords bi-univalentfunctionsFaberpolynomialscoefficientestimatesFekete-SzegőinequalitysubordinationunivalentMaclaurincoefficientsbounded

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 bi-univalent functions, meaning $f$ and its inverse $f^{-1}$ are both univalent in the unit disk, and their coefficient sequences are hard to control because the inverse coefficients depend nonlinearly on the original ones. It defines a broad three-parameter class $H_\Sigma(\tau,\lambda,\delta;\phi)$ by requiring both $f$ and $f^{-1}$ to satisfy a subordination condition, meaning the relevant expressions stay inside the image region of an analytic function $\phi(z)=1+B_1z+B_2z^2+\cdots$ with $B_1>0$. Using Faber polynomial expansions, the paper derives bounds for $|a_2|$, $|a_3|$, and the Fekete–Szegő functional $|a_3-2a_2^2|$ when $B_1\ge|B_2|$, plus a bound for $|a_n|$ with $n\ge4$ when earlier coefficients vanish. If correct, these inequalities give unified control over a wide family and recover several known special cases.

What carries the argument

The engine is the Faber polynomial formula $A_n=\frac{1}{n}K_{n-1}^{-n}(a_2,\dots,a_n)$ for the coefficients of the inverse function; these polynomials express each inverse coefficient as a homogeneous polynomial in the forward coefficients, turning the subordination conditions into explicit identities. The second engine is the Schwarz-function coefficient inequality from Lemma 2.2 and Lemma 2.3, which bounds combinations like $c_2+(B_2/B_1)c_1^2$ by expressions of the form $1-((B_1\pm B_2)/B_1)|c_1|^2$. The equality $c_1=-d_1$ forced by the two subordinations is what lets the two equations combine into the final bounds.

What would settle it

Compare equations (45) and (48) with inequality (10). For $B_2<0$, Lemma 2.3 gives $\le 1-((B_1+B_2)/B_1)|c_1|^2$; for $B_2>0$ it gives $\le 1-((B_1-B_2)/B_1)|c_1|^2$. The paper's equations swap those signs. Checking which sign the lemma actually supplies settles whether the stated $|a_3|$ and $|a_3-2a_2^2|$ bounds are consequences of the cited lemma.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that the Faber polynomial expansion of the inverse function $f^{-1}(w)=w+\sum_{n=2}^{\infty}A_nw^n$ converts the two subordination conditions defining $H_\Sigma(\tau,\lambda,\delta;\phi)$ into algebraic equations linking $a_2$ and $a_3$ with the first coefficients $c_1,c_2,d_1,d_2$ of the two bounded analytic comparison functions (Schwarz functions) $u$ and $v$. Adding and subtracting those equations yields $a_2^2=\frac{B_1\tau}{2(1+2\lambda+6\delta)}[(c_2+\frac{B_2}{B_1}c_1^2)+(d_2+\frac{B_2}{B_1}d_1^2)]$, $a_3=\frac{\tau(B_1c_2+B_2c_1^2)}{1+2\lambda+6\delta}$, and $a_3-2a_2^2=-\frac{\tau(B_1d_2+B_2d_1^2)}{1+2\lambda+6\delta}$. Applying a coefficient inequality for Schwarz functions, the paper claims that for $B_1\ge|B_2|$ the coefficient $|a_2|$ is bounded by $B_1\sqrt{B_1|\tau|}/\sqrt{B_1^2|\tau|(1+2\lambda+6\delta)+(B_1\pm B_2)(1+\lambda+2\delta)^2}$, with the sign of $B_2$ selecting the sign in the denominator, while $|a_3|$ and $|a_3-2a_2^2|$ are bounded by $B_1|\tau|/(1+2\lambda+6\delta)$ in one stated case and by $|B_2\tau|/(1+2\lambda+6\delta)$ in the other. It also proves $|a_n|\le B_1|\tau|/(1+(n-1)(\lambda+n\delta))$ for $n\ge4$ when $a_2=\cdots=a_{n-1}=0$.

Load-bearing premise

The central estimates depend on applying the bounding lemma with the correct sign for the second coefficient of $\phi$; if that sign is mismatched, the case splits and bounds do not follow.

Editorial extensions

If this is right

Setting $\lambda=1$ in Theorem 2.8 yields bounds for the class $\Sigma(\tau,\delta,\phi)$ that the paper lists as Corollary 2.9.
Choosing $\phi(z)=((1+z)/(1-z))^\alpha$ gives the bounds in Corollaries 2.10 and 2.12 for the classes $H_\Sigma(\alpha,\delta)$ and $B_\Sigma(\alpha,\lambda)$.
Choosing $\phi(z)=(1+z)/(1-z)$ and $\tau=1-\gamma$ recovers the bounds in Corollaries 2.11, 2.13, and 2.14 for $H_\Sigma(\gamma,\delta)$, $N_\Sigma(\gamma,\lambda,\delta)$, and $B_\Sigma(\gamma,\lambda)$.
The paper states that these specializations improve the earlier estimates of $|a_3|$ for the corresponding classes.

Reading between the lines

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

The same coefficient-comparison setup could in principle be pushed to $n\ge4$ without the vanishing-coefficient assumption, but the Faber expressions grow quickly and the paper does not do that.
A corrected sign handling would likely reduce the two $B_2$ cases to one unified expression involving $B_1\pm B_2$, which would also clarify where equality might be attained.
The paper supplies no extremal examples, so a natural test is to construct functions in $H_\Sigma(\tau,\lambda,\delta;\phi)$ that attain the claimed bounds.

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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

Quick take: the class HΣ(τ,λ,δ;φ) is a legitimate new parameterization, and Theorem 2.4 is a correct, reasonably useful generalization of known coefficient bounds. But Theorem 2.8, the centerpiece, is not established. The printed |a2| estimate is off by a factor √|τ| from the proof, the case conditions B1+B2≤0 and B1−B2≤0 are impossible under the stated hypothesis B1≥|B2|, and the |a3| argument applies Lemma 2.3 with the sign of B2 reversed. The branch structure collapses once those signs are fixed.

Theorem 2.4 is solid: if a2,…,a_{n−1}=0, then |an| ≤ B1|τ|/(1+(n−1)(λ+nδ)). The algebra checks out, the use of An=−an and Lemma 2.2 is fine, and this genuinely generalizes several published results. That part is worth having.

Theorem 2.8 is another story. For |a2|, substituting c1 from (33) into (36) gives |a2|² ≤ |τ|²B1³ / (B1²|τ|(1+2λ+6δ)+(B1+B2)(1+λ+2δ)²), so the square root is B1^{3/2}|τ| over the root of the denominator. The statement has B1√(B1|τ|), which is off by √|τ|. Worse, the proof itself uses B1+B2≥0 and B1−B2≥0, which are implied by B1≥|B2|; the theorem states the opposite inequalities, making the stated cases vacuous. So the |a2| bound as printed is not a theorem, though it is close to a correct one.

The |a3| and Fekete–Szegö estimates are more seriously flawed. Lemma 2.3 for η=B2/B1<0 gives |c2+ηc1²| ≤ 1−((B1+B2)/B1)|c1|², but (45) uses (B1−B2)/B1. For η>0, the roles are swapped in (48). This is not a minor typo: the subsequent maximization over |c1| is done with the wrong coefficient, so the cases B1−B2<0 and B1+B2<0 are spurious. Under the hypothesis B1≥|B2|, both correct bracket terms are nonnegative, so the maximum is at |c1|=0 and you just get |a3| ≤ B1|τ|/(1+2λ+6δ). The second branch B1<|B2| contradicts the hypothesis, and the intermediate bound in (47) is negative when B2<0. So the advertised B2-dependent bounds for |a3| and |a3−2a2²| do not survive; the corrected theorem would be a simpler, less exciting statement.

On the positive side, the paper is not circular and the citations are honest. The corollaries show real connections to the existing literature, though the claimed improvements of earlier |a3| estimates rest on the broken Theorem 2.8 and need re-examination. The paper would be useful to researchers in bi-univalent coefficient problems as a way to see the class and to find the correct Theorem 2.4. But as is, the main theorem is not trustworthy.

For peer review: I would send it out, because the errors are specific, partly fixable, and the paper has a correct kernel. My expectation is that a referee would require major revision or, if Theorem 2.8 cannot be rescued in its advertised generality, reject the current version.

Referee Report

4 major / 4 minor

Summary. The paper defines a subclass HΣ(τ,λ,δ;φ) of bi-univalent functions through subordination conditions on a weighted expression in f and its inverse, and uses Faber polynomial expansions to derive coefficient estimates. Theorem 2.4 gives a general bound for |a_n| under the vanishing-coefficient hypothesis, and Theorem 2.8 claims bounds for |a2|, |a3|, and the Fekete-Szegö functional |a3−2a2^2|. Corollaries 2.9–2.14 specialize Theorem 2.8 to previously studied classes. The central claim of the paper is Theorem 2.8, and the corollaries all depend on it.

Significance. If Theorem 2.8 were correct, the paper would unify and improve several earlier coefficient estimates for bi-univalent function classes, and the Faber polynomial framework for the general coefficient bound would be a useful contribution. Theorem 2.4 is straightforward and appears sound. However, the proof of Theorem 2.8 contains algebraic and logical errors that affect exactly the quantities the theorem is meant to bound: the |a2| bound in (25) does not match the proof's algebra, the case conditions are incompatible with the stated hypothesis, and the application of Lemma 2.3 reverses the signs in the |a3| estimates. Since the advertised improvements are corollaries of this theorem, the central contribution is not established as printed.

major comments (4)

[Theorem 2.8, Eq. (25) and proof Eqs. (37)–(38)] The |a2| estimate stated in (25) does not follow from the proof. From (37) the proof obtains |a2|^2 ≤ |τ|^2 B1^3 / [B1^2|τ|(1+2λ+6δ)+(B1+B2)(1+λ+2δ)^2], whose square root is |a2| ≤ |τ| B1^{3/2} / sqrt(D). The numerator printed in (25) is B1 sqrt(B1|τ|), a factor sqrt(|τ|) smaller. Thus the displayed inequality (25) is not the conclusion of the derivation given in (36)–(38).
[Theorem 2.8, hypothesis and case conditions in (25)] The case split in (25) is incompatible with the standing hypothesis B1 ≥ |B2|. Since B1 > 0, the condition B2 < 0 and B1+B2 ≤ 0 can hold only when B1+B2 = 0, and the condition B2 > 0 and B1−B2 ≤ 0 can hold only when B1−B2 = 0. Hence the two branches give no |a2| bound for generic parameter values. Moreover, the proof itself uses the opposite inequalities: (36) says 'B2 < 0 (η = B2/B1 < 0, B1 + B2 ≥ 0)' and (39) says 'B2 > 0 (η = B2/B1 > 0, B1 − B2 ≥ 0)', so the proof does not even address the cases stated in the theorem.
[Theorem 2.8, proof of (26) using Lemma 2.3, Eqs. (45) and (48)] Lemma 2.3 as stated in (10) is applied with reversed signs. For B2 < 0, η = B2/B1 < 0, and inequality (10) gives 1 − (1+η)|c1|^2 = 1 − ((B1+B2)/B1)|c1|^2, but equation (45) uses 1 − ((B1−B2)/B1)|c1|^2. For B2 > 0, inequality (10) gives 1 − ((B1−B2)/B1)|c1|^2, but equation (48) uses 1 − ((B1+B2)/B1)|c1|^2. These sign errors control the choice |c1| = 0 versus |c1| = 1 in (46)–(50), so the |a3| bound and its case split (26) are not derived.
[Theorem 2.8, Eqs. (26)–(27) and Eq. (47)] The case structure collapses under the theorem's own hypotheses. The branch B1 < |B2| in (26) and (27) contradicts the standing assumption B1 ≥ |B2| and is therefore vacuous. In addition, equation (47) claims |a3| ≤ B2|τ|/(1+2λ+6δ) when B2 < 0, which cannot be true because the right-hand side is negative; this is a direct symptom of the sign error in (45). A corrected application of Lemma 2.3 would remove the B2-dependent branches rather than merely change their bounds.

minor comments (4)

[Definition 2.1, Eq. (8)] In the subordination condition for the inverse function, the term δz g1''(w) should be δw g1''(w); as written, the variable z is used in an expression in w.
[Section 2.1, before Corollary 2.6] The sentence 'Let us put λ=1 in Corollary 2.6' appears to refer to Corollary 2.5, since Corollary 2.6 is stated afterward.
[Lemma 2.3 and Section 2.1] The symbol φ is used both for the superordinating function in (2) and for the Schwarz function in the derivation of inequality (10), which is confusing; a different letter for the Schwarz function would improve readability.
[Throughout] The manuscript contains many typographical and notational glitches in displayed formulas, for example the reference to 'inequality (41)' in the sentence after (40), and the denominator notation in (25) that should read B1^2. A careful proofread is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the coefficient estimates are derived from externally cited lemmas and the new class is a generalization whose corollaries recover earlier results.

full rationale

The derivation chain in Theorems 2.4 and 2.8 is self-contained: the class HΣ(τ,λ,δ;ϕ) is defined by subordination, the Faber-polynomial coefficient identities (19)-(20) are derived in the paper from that definition, and the estimates then follow by applying the externally cited Schwarz lemma (Lemma 2.2, [18]) and a Fekete-Szegő-type inequality (Lemma 2.3, [7]). Nothing is fitted to data, and no target coefficient bound is assumed as an input. The only self-citation is reference [9] in the introductory list of recent papers on bi-univalent functions; it is not used to justify any result and is therefore not load-bearing. The corollaries reproduce earlier results by other authors (e.g., [22], [11], [10], [6], [21]) as special cases, which is a consistency check rather than a circular derivation. Possible mathematical errors in the application of Lemma 2.3, as flagged by skeptical review, are correctness concerns, not circularity, and they do not raise the circularity score.

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

The work introduces no fitted constants or new physical entities. It relies on standard Faber polynomial machinery and two cited lemmas. The fragile external input is Lemma 2.3, which is unproved and misapplied in the proof.

assumptions (5)

standard math Faber polynomial expansion of the inverse function f^{-1} (equations (4)-(6))
Standard expansion from Refs. [1,2]; the paper adopts it without proof.
standard math Schwarz coefficient bound |c_n| ≤ 1 from Lemma 2.2
Classical result [18] invoked for Theorems 2.4 and 2.8.
domain assumption Carathéodory-type inequality (10) for Schwarz coefficients from Lemma 2.3
Cited from [7] without proof; used centrally in Theorem 2.8 and applied with signs inconsistent with the lemma.
domain assumption Subordination representation via Schwarz functions (14)-(15)
Defines the class HΣ; assumes existence of such functions u,v for each member.
domain assumption Parameter conditions λ ≥ 1, τ ∈ C*, 0 ≤ δ ≤ 1, B1 > 0
Domain restrictions from Definition 2.1 and (2); they are assumptions, not derived.

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Pith. "Pith review of Faber polynomial coefficient estimation of subclass of bi-subordinate univalent functions." pith.science (2026). https://pith.science/paper/NF5OUR3O

@misc{pith2026190807349,
  author       = {Pith},
  title        = {Pith review of: Faber polynomial coefficient estimation of subclass of bi-subordinate univalent functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NF5OUR3O}},
  note         = {Machine review of arXiv:1908.07349}
}

abstract

In this paper, a comprehensive subclass of bi-univalent functions class are introduced and investigated. Using the Faber polynomials, estimation of the coefficients $|a_n|$ and certain Fekete-Szeg\"{o} inequality of Maclaurin expansion of functions in this subclass are concluded. Finally, some earlier results are pointed out and improved.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Airault and A

H. Airault and A. Bouali, Diﬀerential calculus on the Faber polynomials , Bull. Sci. Math., 130(3), 2006, 179-222

work page 2006

[2] [2]

Airault, Remarks on Faber Polynomials , Int

H. Airault, Remarks on Faber Polynomials , Int. Math. Forum, 3(9), 2008, 449-456. 10

work page 2008

[3] [3]

Ali, S.K

R.M. Ali, S.K. Lee, V . Ravichandran and S. Supramanian, Coeﬃcient estimates for bi-univalent Ma- Minda starlike and convex functions , Appl.Math. Lett., 25, 2012, 344-351

work page 2012

[4] [4]

D. A. Brannan and J. G. Clunie, Aspects of Contemporary Complex Analysis , (Proceedings of the NATO Advanced Study Institute held at the University of Durh am, Durham; July 1 20, 1979), Academic Press, New York and London, 1980

work page 1979

[5] [5]

Brannan and T.S

D.A. Brannan and T.S. Taha, On some classes of bi-univalent functions , Studia Univ . Babes ¸-Bolyai Math., 31(2), 1986, 70-77

work page 1986

[6] [6]

Bulut, Faber polynomial coe ﬃcient estimates for a subclass of analytic bi-univalent functi ons, Filomat, 30(6), 2016, 1567-1575

S. Bulut, Faber polynomial coe ﬃcient estimates for a subclass of analytic bi-univalent functi ons, Filomat, 30(6), 2016, 1567-1575

work page 2016

[7] [7]

Deniz, J.M

E. Deniz, J.M. Jahangiri, S.G. Hamidi and S.K. Kina, Faber polynomial coe ﬃcients for generalized bi-subordinate functions of complex order , J. Math. Inequal., 12(3), 2018, 645-653

work page 2018

[8] [8]

Duren, Univalent Functions

P .L. Duren, Univalent Functions. In: Grundlehren der Mathematischen Wis senschaften, New York, Berlin, Heidelberg and Tokyo, Springer-V erlag, 1983

work page 1983