Some estimation about Tayler-Maclaurin coefficients of generalized subclasses of bi-univalent functions

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

REVIEW 3 major objections 4 minor 23 references

Some estimation about Tayler-Maclaurin coefficients of generalized subclasses of bi-univalent functions

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves explicit upper bounds for the second and third Taylor-Maclaurin coefficients of two new generalized subclasses of bi-univalent functions.

desk verdict Routine parameter-extension paper whose second |a3| bound in Theorem 2.1 doesn't survive a careful reading: a dropped term and a false inequality sink it. read the letter →

arxiv 1908.08042 v1 pith:CE2H5AWR submitted 2019-08-21 math.CV

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

classification math.CV MSC 30C4530C50

keywords bi-univalentfunctionscoefficientboundsTaylor-MaclaurincoefficientsstarlikeconvexCarathéodorylemmaunivalentanalytic

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

The paper introduces two broad families of bi-univalent functions, $S^\alpha_\Sigma(\tau,\delta,\lambda,\gamma)$ and $S_\Sigma(\tau,\delta,\mu,\lambda,\gamma;\beta)$, defined by angular or real-part conditions on a differential expression built from $f$, $zf'$, and $z^2f''$. For both families it proves explicit upper bounds for $|a_2|$ and $|a_3|$ in terms of the five parameters and a composite denominator $\Omega$. The bounds specialize to recover many previously known coefficient estimates and, in several corollaries, sharpen them. Coefficient bounds are the basic quantitative measure of how far a bi-univalent function can deviate from the identity map, so a unified family with explicit bounds condenses many earlier results into one theorem.

What carries the argument

The central object is the coefficient-comparison identity that links the class conditions to the Carathéodory functions $h_1,h_2$. Writing the defining angular conditions as $(h_1(z))^\alpha$ and $(h_2(w))^\alpha$ and comparing coefficients yields equations (2.6)-(2.9), which express $a_2$ and $a_3$ in terms of $p_1,p_2,q_1,q_2$. The composite parameter $\Omega$ in (2.3) is the effective denominator that carries all parameter dependence once $a_2^2$ is eliminated; the final bounds come from applying the Carathéodory bound $|c_n|\le2$ to the resulting expressions.

What would settle it

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.1: every $f\in S^\alpha_\Sigma(\tau,\delta,\lambda,\gamma)$ obeys $$|a_2|\le \frac{2\$\alpha$|\tau|}{\sqrt{|2\$\alpha$\tau\$\Omega$+(1-\$\alpha$)(1+\delta+2\mu-\$\lambda$-\gamma\$\lambda$)^2|}}$$ and $$|a_3|\le \min\left\{\frac{4\$alpha^{2}$|\tau|^2}{(1+\delta+2\mu-\$\lambda$-\gamma\$\lambda$)^2}+\frac{2\$\alpha$|\tau|}{|1+2\delta+6\mu-\$\lambda$-2\gamma\$\lambda$|},\frac{2\$\alpha$|\tau|}{|\$\Omega$|}\right\},$$ with $\Omega$ defined by (2.3). Theorem 3.1 gives the analogous bounds for the real-part class $S_\Sigma(\tau,\delta,\mu,\lambda,\gamma;\beta)$, with $\alpha$ replaced by $1-\beta$. The paper derives both theorems by representing the class conditions as powers of Carathéodory functions, comparing coefficients with those of $f$ and its inverse, and applying the classical estimate $|c_n|\le2$.

Load-bearing premise

Editorial extensions

If this is right

For the parameter choice $\delta=1$, Theorems 2.1 and 3.1 reduce to modified coefficient estimates for the classes $H_\Sigma(\tau,\mu,\lambda,\gamma;\alpha)$ and $H_\Sigma(\tau,\mu,\lambda,\gamma;\beta)$ studied earlier.
Other parameter specializations recover the coefficient bounds for the subclasses $N_\Sigma$, $G_\Sigma$, $M_\Sigma$, $B_\Sigma$, and the $H$-type classes listed in Remarks 2-10.
In Corollaries 4.10-4.15 the paper's $|a_3|$ estimates improve on three earlier sets of bounds.
The two theorems provide a single parameter-dependent formula from which many previously separate estimates follow as corollaries.

Reading between the lines

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

A direct numerical scan over the admissible cube $(\delta,\mu,\lambda,\gamma)\in[0,1]^4$ would show which branch of the $|a_3|$ minimum is active in each region; the paper does not chart this transition.
The same coefficient-comparison method should extend to $|a_4|$ in these classes, although the paper stops at $|a_3|$ and the general bi-univalent coefficient problem for $n\ge4$ remains open.
Because the two defining conditions are symmetric under $f\leftrightarrow f^{-1}$, sharpness examples for the $|a_2|$ bound would automatically constrain the inverse side as well.

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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

Short version: the |a2| bound and the first |a3| bound in Theorem 2.1 are fine; the second |a3| bound is not. The proof drops a nonzero α(α−1)p1² term when substituting (2.12) into (2.16), and the triangle inequality used to finish the estimate is false for allowed parameter values. The same trouble appears in the second |a3| bound of Theorem 3.1. This is load-bearing, not a typo.

The class SαΣ(τ,δ,λ,γ) is a genuine five-parameter unification of several known bi-univalent classes, and the |a2| estimate follows the standard Carathéodory template correctly. The first |a3| estimate also comes out of the usual computation. So the paper is not empty; it's a competent but routine extension of Srivastava et al. [22].

The algebra: from (2.7)–(2.9) you get (2.12); substituting into (2.16) yields a3 = (ατ/2)[p2(1/Ω+1/B)+q2(1/Ω−1/B)] + ατ(α−1)p1²/(2Ω). Equation (2.17) omits the last term, which is generically nonzero for 0<α<1 and is uncontrolled. Independently, the proof needs |1/Ω+1/B| + |1/Ω−1/B| ≤ 2/|Ω|. That fails: take δ=µ=0, γ=1/2, λ=0.49, then B=0.02, Ω≈−0.1748, so the left side is 100 and the right side about 11.4. The second |a3| bound in Theorem 3.1 relies on the same false inequality.

The paper also has many typos and notation slips, and Remark 11 claims improvements without giving the earlier bounds, so a reader can't check the comparisons.

Who is this for? Someone building a reference list of coefficient bounds for bi-univalent subclasses will find the |a2| result and the first |a3| bound useful. But the main theorem as stated is not established, and the open |a4| problem is untouched. I'd send it to review—there's enough substance that a referee should catch the algebraic slip and the authors might repair it—but it shouldn't be accepted in its present form.

Referee Report

3 major / 4 minor

Summary. The paper introduces two families of normalized analytic bi-univalent functions, SαΣ(τ,δ,λ,γ) and SΣ(τ,δ,µ,λ,γ;β), defined by differential-operator-type subordination conditions involving parameters τ, δ, µ, λ, γ. The main results are Theorems 2.1 and 3.1, which claim bounds for the second and third Taylor–Maclaurin coefficients of functions in these classes, with the |a3| bounds stated as a minimum of two branches. The proofs use the Carathéodory coefficient estimates for functions with positive real part and compare coefficients of f and f^{-1}. Section 4 derives corollaries that specialize the parameters to previously studied subclasses.

Significance. If the bounds were valid, the paper would provide a fairly general coefficient-estimate framework for bi-univalent functions and would improve several published results. Credit is due for the parts that are correct: the derivation of the |a2| bound and of the first branch of the |a3| bound in Theorem 2.1 follows the standard Carathéodory-lemma method and is carried out correctly. However, the second branch of the |a3| bound in Theorem 2.1, which is essential to the stated minimum, relies on an algebraic omission and on a triangle inequality that is false for admissible parameters. The same defective inequality is used in Theorem 3.1, and the corollaries inherit the invalid branch whenever it is the smaller one. The central claims of the paper are therefore not established as stated.

major comments (3)

[Section 2, Eq. (2.17)] The displayed expression for a3 is obtained from (2.12) and (2.16) only if the term ατ(α−1)p1²/(2Ω) is dropped. Because p1 = −q1, equation (2.12) gives 2Ωa2²/τ = α(p2+q2)+α(α−1)p1². Substituting this into (2.16) yields a3 = ατ/2 [p2(1/Ω+1/B)+q2(1/Ω−1/B)] + ατ(α−1)p1²/(2Ω). For 0 < α < 1 this residual term does not vanish and is not controlled anywhere in the proof, so the claimed bound |a3| ≤ 2α|τ|/|Ω| does not follow.
[Section 2, proof of Theorem 2.1] The second bound in (2.2) requires the inequality |1/Ω+1/B| + |1/Ω−1/B| ≤ 2/|Ω|, where B = 1+2δ+6µ−λ−2γλ. This inequality is false for allowed parameters: for δ=µ=0, γ=1/2, λ=0.49 one has B=0.02 and Ω=−0.174775, so the left side equals 100 while 2/|Ω| ≈ 11.44. Therefore the min in (2.2) is not justified by the given argument.
[Section 3, Theorem 3.1, second branch of (3.2)] The derivation of |a3| ≤ 2(1−β)|τ|/|Ω| from (3.11) and (3.14) again uses the inequality |1/Ω+1/B| + |1/Ω−1/B| ≤ 2/|Ω|. With the same admissible parameter values this inequality fails, so the second branch of (3.2) is unproved. Consequently the corollaries in Section 4 that select this branch are not supported.

minor comments (4)

[Definition 1.1 and Theorem 2.1] The class SαΣ(τ,δ,λ,γ) is not consistently defined: Definition 1.1 lists a parameter µ in the defining inequalities, but the notation omits µ, and Theorem 2.1 uses the same notation even though Ω depends on µ.
[Throughout Section 2, Eq. (2.1) and (2.2)] The cross-references are incorrect: the proof refers to “the desired estimate of a2 given by (4.1)” where (2.1) is meant, and the analogous reference “(4.2)” in Section 3 should point to (3.1) and (3.2).
[Throughout] There are numerous typographical errors, including “Tayler” in the title, “boss sides” for “both sides”, and the rendering “/g1” for the inverse function.
[Remark 2, items 4 and 5] Items 4 and 5 both define a class denoted BΣ(α,λ) with different parameter specializations; if both notations are intended, they should be disambiguated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the coefficient bounds are derived from the class definition and the standard Carathéodory lemma, with no fitted inputs or self-citation chain.

full rationale

The paper's Theorem 2.1 derives bounds on |a2| and |a3| from the defining inequalities of the class SαΣ(τ,δ,λ,γ) by introducing Carathéodory functions h1,h2, comparing coefficients, and applying the classical coefficient estimate |cn| ≤ 2 (Lemma 1.3, cited to Duren [9]). The derivation is self-contained algebra after this standard lemma: equations (2.6)–(2.17) manipulate the coefficient identities, and the final bounds are obtained by substituting the Carathéodory coefficient bounds. No parameter is fitted to the target bound; the class parameters τ,δ,µ,λ,γ appear as variables in the statement, not as fitted constants. There is no self-citation that carries the argument: the cited external results are the classical Carathéodory lemma and prior definitions of special cases, neither of which is used to force the theorem's conclusion. The potential mathematical errors flagged by the reader (e.g., the omitted α(α−1)p1² term in passing from (2.12) to (2.17), and the possibly false triangle inequality used after (2.17)) are correctness or validity concerns about the proof, not circularity: the proof does not assume the conclusion or define the class in terms of the bound. Thus no circular reduction is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No new entities or fitted constants. The derivation builds on standard lemmas and the series expansion of the inverse function.

assumptions (3)

standard math Carathéodory lemma: if h(z)=1+c1z+c2z^2+... has positive real part in U, then |cn| ≤ 2, sharp.
Stated as Lemma 1.3 and used throughout Sections 2 and 3 to bound p_n and q_n.
domain assumption Any function satisfying |arg w| < απ/2 can be written as h(z)^α for a Carathéodory function h.
Used in Theorem 2.1 proof to set up equations (2.4)-(2.5). Standard in this literature.
standard math The inverse function g(w) = f^{-1}(w) has the series g(w) = w - a2 w^2 + (2a2^2 - a3) w^3 + ..., valid in the unit disc.
Equation (1.4), derived from the Lagrange inversion formula.

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Pith. "Pith review of Some estimation about Tayler-Maclaurin coefficients of generalized subclasses of bi-univalent functions." pith.science (2026). https://pith.science/paper/CE2H5AWR

@misc{pith2026190808042,
  author       = {Pith},
  title        = {Pith review of: Some estimation about Tayler-Maclaurin coefficients of generalized subclasses of bi-univalent functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CE2H5AWR}},
  note         = {Machine review of arXiv:1908.08042}
}

read the original abstract

Our objective in this paper is to introduce and investigate comprehensive-constructed subclasses of normalized analytic and bi-univalent functions on the unit open disc. Bounds for the second and third Tayler-Maclaurin coefficients of functions belonging to this subclasses were investigated. Furthermore, some improvement and connections to some of the previous known results are also pointed out.

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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] [22]

& Ghanim, F.(2016), Coe ﬃcient estimates for some general subclasses of analytic and bi-univalent functions, Afr

Srivastava, H.M., Gaboury , S. & Ghanim, F.(2016), Coe ﬃcient estimates for some general subclasses of analytic and bi-univalent functions, Afr. Mat ., 28(5-6), 693-706

work page 2016

[2] [1]

Altinkaya, S ¸ . & Yalc ¸in, S.(2017), Estimates on coeﬃcients of a general subclass of bi-univalent functions associated with symmetric q-derivative operato r by means of the Chebyshev poly- nomials, Asia Paciﬁc Journal of Mathematics, 4(2), 90-99

work page 2017

[3] [2]

& Darus, M.(2017 ), On H 3(p) Hankel determinant for certain subclass of p-valent functions, Ital

Amourah, A.A., Yousef, F., Al-Hawary , T. & Darus, M.(2017 ), On H 3(p) Hankel determinant for certain subclass of p-valent functions, Ital. J. Pure Ap pl. Math., 37, 611-618

work page 2017

[4] [3]

& Yousef, F.(2018), Coe ﬃcients estimates for certain classes of analytic functions of complex order, Afr

Al-Hawary , T., Frasin, B.A. & Yousef, F.(2018), Coe ﬃcients estimates for certain classes of analytic functions of complex order, Afr. Mat., 29(7-8), 126 5-1271

work page 2018

[5] [4]

& Kirwan, W.E.(1970), Coe ﬃcient estimates for a class of star-like functions, Canad

Brannan, D.A., Clunie, J.G. & Kirwan, W.E.(1970), Coe ﬃcient estimates for a class of star-like functions, Canad. J. Math., 22, 476-485. 9

work page 1970

[6] [5]

Brannan, D.A. & Clunie, J.G.(1980), Aspects of contempo rary complex analysis (Proceedings of the NATO Advanced Study Institute held at the University of Durham, Durham, Academic Press, New York and London)

work page 1980

[7] [6]

& Tan, D.L.(1986), On some classes of bi-un ivalent functions, Stud

Brannan, D.A. & Tan, D.L.(1986), On some classes of bi-un ivalent functions, Stud. Univ . Babes- Bolyai Math., 31(2), 70-77

work page 1986

[8] [7]

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

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

work page 2016