Upper bound of second Hankel determinant of subclass of bi-univalent functions defined by subordination

Alaa H. El-Qadeem; Mohamed A. Mamon

REVIEW 2 major objections 7 minor 22 references

Upper bound of second Hankel determinant of subclass of bi-univalent functions defined by subordination

T0 review · 2 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper establishes a single explicit upper bound for the second Hankel determinant across a new bi-univalent class defined by subordination, which yields estimates for seven known subclasses and corrects two earlier ones.

desk verdict A likely correct but under-proved unified Hankel bound for a new bi-univalent class; the missing maximization calculation is fixable and the theorem seems sound. read the letter →

arxiv 1908.07350 v1 pith:KP6SEKXV submitted 2019-08-08 math.CV

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

classification math.CV MSC 30C4530C5030C55

keywords analyticfunctionsunivalentbi-univalentsubordinationHankeldeterminantsecondcoefficientestimatesSchwarzfunction

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 introduces a class of bi-univalent functions HΣ(τ,λ,δ;φ): functions f(z)=z+a2z2+a3z3+... whose normalized differential expression (1−λ)f(z)/z+λf′(z)+δzf″(z) is subordinate to a univalent function φ(z)=1+B1z+B2z2+B3z3+... with positive real part, and the same expression for the inverse f−1 is subordinate to φ. The main theorem claims that for λ≥1, 0≤δ≤1, and τ∈C∖{0}, every such f satisfies |a2a4−a32|≤B1|τ|2(P+Q+R), with explicit P,Q,R depending on the parameters and on B1,B2,B3. This matters because the class is a common umbrella for at least seven earlier subclasses, so the theorem yields their second-Hankel bounds as corollaries. The authors also state that two earlier published estimates were obtained by miscalculation and present corrected versions.

What carries the argument

The load-bearing object is the function class HΣ(τ,λ,δ;φ), defined by the two subordination conditions (1.4)–(1.5) applied to f and f−1. The argument runs through three standard tools: the coefficient equations obtained by expanding φ(ω(z)) and comparing powers; Lemma 1.2's parametrization of the Schwarz coefficients c2 and c3 by |c1|≤1, |x|≤1, |ξ|≤1, which turns the determinant into a bound B1|τ|2F(ν,μ) with F=T1+(ν+μ)T2+(ν2+μ2)T3+(ν+μ)2T4; and the maximization of F over [0,1]2. The key move is showing the maximum sits at the corner F(1,1), after which the remaining one-variable maximization in c∈[0,1] is a quadratic Pt2+Qt+R.

What would settle it

A reader can settle the proof: pick any admissible parameter tuple, compute T3 and T4, and check whether T3+2T4≥0 and T3+T4≥0. Finding one admissible tuple where either inequality fails and where F(ν,μ) has a maximum larger than F(1,1) would expose the gap; to refute the theorem itself one would then need a member of HΣ with |a2a4−a32| larger than B1|τ|2(P+Q+R), which the explicit formula for the determinant in terms of c1,x,y,ξ,η makes searchable.

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Extended reading notes

Core claim

On its own terms, the paper claims the following. Let λ≥1, 0≤δ≤1, τ∈C∖{0}, and let φ(z)=1+B1z+B2z2+B3z3+... have positive real part with B1>0 and B2,B3 real. If f and its inverse f−1 both make the differential expression (1−λ)f(z)/z+λf′(z)+δzf″(z) subordinate to φ, then |H2(2)|=|a2a4−a32|≤B1|τ|2(P+Q+R), where P,Q,R are the explicit nonnegative expressions in Eq. (2.2). The proof compares the series coefficients of the two subordination conditions, uses the Schwarz-function parametrization c2=(1−c12)x and c3=(1−c12)(1−|x|2)ξ−c1(1−c12)x2, and reduces the problem to maximizing a polynomial F(ν,μ) on the unit square; the claimed maximum occurs at the corner (ν,μ)=(1,1) and then at c=1. The paper presents this as a unified result whose parameter specializations recover or correct earlier bounds in the literature.

Load-bearing premise

The proof depends on an assertion, made without showing the calculation, that a certain algebraic expression in the parameters is nonnegative for all allowed choices. If that assertion fails, the maximum of the auxiliary function need not occur at the corner, and the stated bound would not follow.

Editorial extensions

If this is right

For every specialization listed in Remark 1, Theorem 2.1 gives a ready-made upper bound on |a2a4−a32|, so the seven subclasses inherit the result without separate proofs.
Corollaries 3.4 and 3.7 supply corrected versions of the two estimates from [7] that the paper identifies as miscalculated.
The structure of the bound is a positive quadratic in c∈[0,1] evaluated at c=1, so the estimate is explicit and directly computable once φ's coefficients B1,B2,B3 are known.
Any future class that can be written in the form HΣ(τ,λ,δ;φ) with the stated parameter ranges automatically satisfies the same Hankel bound.

Reading between the lines

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

The paper leaves implicit whether the same corner-maximization pattern persists for higher-order Hankel determinants; the coefficient-comparison strategy is general, but the auxiliary maximization would become higher-dimensional and the unproved inequality would need to be supplied.
A reader who wants to apply Theorem 2.1 outside the stated parameter ranges would first need to verify the T3+2T4≥0 assertion numerically or symbolically; a single admissible tuple where it fails and the maximum of F exceeds F(1,1) would expose the gap.
The maximizing configuration (c,ν,μ)=(1,1,1) suggests that, if the bound is sharp, extremal behaviour is governed by Schwarz functions and auxiliary variables on the unit circle, which could guide numerical searches for extremal functions in HΣ.

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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: this is a modest but legitimate contribution to the bi-univalent/Hankel determinant industry. The class HΣ(τ,λ,δ;φ) is new and the main theorem genuinely specializes to at least seven published classes, so there is real unifying content. The proof is standard—subordination, Schwarz-function parametrization, then a two-variable maximization—and the coefficient algebra checks out.

The load-bearing spot is the maximization of F(ν,μ) on [0,1]^2. The paper asserts T3+2T4 ≥ 0 without proof, then later assumes T3+T4 ≥ 0 to conclude the maximum occurs at the corner. These inequalities are essential: if they failed for admissible parameters, the claimed bound could be too small. I did the substitution with A=1+λ+2δ, C=1+2λ+6δ, D=1+3λ+12δ, and the inequalities do hold for λ≥1, 0≤δ≤1; so the theorem is very likely correct. But the proof as written omits the calculation in exactly the step that makes the bound valid. A referee should ask for it.

Also: the inverse-coefficient formulas in (1.2) and (2.14) have '5a3^2' where it should be '5a2^3', and (2.20) has a typo in a power of τ. These are cosmetic but should be cleaned up. More substantive: Remark 2 claims to correct two earlier estimates, but gives no indication of what the earlier authors wrote, so the correction claim is unverifiable as stated.

No circularity, no fitted parameters. The result is what it claims to be, just under-supported in one place. I would send it to a referee—this is a legitimate paper that would benefit from a routine revision. Not for a general reading group, but for anyone working on Hankel determinants of bi-univalent classes it is worth knowing.

Referee Report

2 major / 7 minor

Summary. The paper introduces a class HΣ(τ,λ,δ;φ) of bi-univalent functions defined by two subordination conditions, one for f and one for its inverse, with parameters λ≥1, 0≤δ≤1, τ∈C∖{0}, and a Ma-Minda type function φ(z)=1+B1z+B2z^2+B3z^3+… . Theorem 2.1 claims an upper bound for the second Hankel determinant |a2a4−a3^2| of the form B1|τ|^2(P+Q+R), with explicit nonnegative expressions P,Q,R. The proof expresses a2,a3,a4 in terms of the coefficients of two Schwarz functions, applies the cited representation lemma for Schwarz functions, and reduces the problem to maximizing a function F(ν,μ) on the unit square. Several corollaries specialize the theorem to previously studied classes, and two corollaries are claimed to correct earlier results of Çağlar et al.

Significance. If Theorem 2.1 is correct, it provides a reasonably general unified upper bound for the second Hankel determinant over a wide subclass of bi-univalent functions, subsuming at least seven known classes through parameter and φ specializations. The coefficient algebra is mostly explicit and checkable, no target bound is assumed, and the final parameter dependence is concrete rather than fitted. The main weakness is not the concept but the execution: the maximization step contains unproved sign assertions that are load-bearing, and several equations and corollaries contain algebraic or notational typos. With those repaired, the paper could be a useful contribution to the bi-univalent coefficient-estimates literature.

major comments (2)

[§2, after Eq. (2.31) and Eqs. (2.32)–(2.40)] The proof of Theorem 2.1 hinges on showing that the maximum of F(ν,μ) on [0,1]^2 occurs at (1,1). The text asserts 'with some calculations we can ensure that T3+2T4≥0' and later concludes that the maximum occurs 'only at T3+T4≥0', but no calculation is supplied. These sign conditions are load-bearing: T3+2T4≥0 is used to rule out interior maxima via the Hessian; T3+T4≥0 is used to make the boundary functions Φ and Ψ increasing; and Φ(1)≤Ψ(1) is equivalent to T2+T3+3T4≥0. If any of these failed for an admissible parameter choice, the final bound B1|τ|^2(P+Q+R) could be too small. In the T3+T4<0 alternatives, the proof only bounds the maximum by T1+T2 or T1+2T2+T3+3T4 and does not compare these bounds with Φ(1) or Ψ(1), so the statement that the maximum occurs only when T3+T4≥0 is not established. The authors must replace the assertion with an explicit verification; the inequalities are in fact true for the stated parameter range, but they need to be shown or proved in a lemma.
[§2, Eq. (2.14)] The coefficient multiplying (5a2^3−5a2a3+a4) is written as (1+λ+2δ)/τ, but from the inverse-series formula (1.2) and the expansion in (2.4) it should be (1+3λ+12δ)/τ. As printed, subtracting (2.14) from (2.13) does not produce Eq. (2.18). The later derivation appears to use the corrected coefficient, so the error is repairable, but the proof cannot be followed as written until Eq. (2.14) is fixed and the subsequent algebra is reconciled.

minor comments (7)

[§2, Eq. (2.20)] The first term in the displayed expression for the Hankel determinant should contain τ^3, not τ^2, as the later bound in Eq. (2.24) confirms. Also, the sentence 'By Using equations (2.16, 2.17, 2.20)' should refer to Eq. (2.19).
[§2, Eq. (2.2) and Eq. (2.24)] The notation rendered as 'B3 1τ2' in the definition of P is ambiguous; it should be written as B_1^3 τ^2. The same notational issue appears in Eq. (2.24).
[§2, Eq. (2.31)] Since T3<0, the Hessian product 4T3(T3+2T4) is strictly negative only when T3+2T4>0; if T3+2T4=0 the Hessian criterion is inconclusive. The strict version is what is needed, so the statement should say T3+2T4>0, or else handle the zero case separately.
[§3, Corollary 3.7] Corollary 3.7 states that f∈Nσ(β), but the displayed bound is written in terms of α throughout; the prefactor 2(1−α)^2 should be 2(1−β)^2 and the remaining α's should be β's.
[§3, Corollary 3.3] Specializing the main theorem with τ=1, λ=1, δ=β and φ(z)=((1+z)/(1−z))^α gives the second term inside the absolute value as α^3/(2(1+β)^4), not α^3/(4(1+β)^4). Please check this corollary against the theorem.
[§3, Corollary 3.6] The analogous specialization for HΣ(α,δ) gives the second term in the absolute value as (1−α)^2/(2(1+δ)^4), not (1−α)^2/(2(1+δ)^2). Please verify the displayed formula.
[Throughout] There are numerous small typographical errors: 'bi-univalant', 'Schawrz' in Lemma 1.2, 'univalant', and the use of 'iﬀ'. These should be corrected in a careful revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Hankel bound follows from coefficient comparisons and standard Schwarz lemmas; the only gap is an omitted algebraic check in the maximization step.

full rationale

The derivation of Theorem 2.1 is self-contained in the relevant sense. The class HΣ(τ,λ,δ;φ) is defined directly by subordination conditions (1.4)–(1.5), and the proof obtains the coefficient relations (2.9)–(2.18) by comparing Taylor coefficients of φ(u(z)) and φ(v(w)) with the class-defining expressions. No parameter is fitted to the target Hankel determinant; the only external inputs are standard Schwarz-function coefficient bounds (Lemma 1.2 from Kanas et al. and |cn|≤1). The final bound B1|τ|²(P+Q+R) is obtained by bounding |H2(2)| with an explicit function F(ν,μ), then maximizing over [0,1]² and [0,1] using nonnegativity of the displayed coefficients P,Q,R. The paper does not invoke any prior result of the present authors, and no uniqueness theorem or ansatz is smuggled in by citation. The proof's unproved 'with some calculations we can ensure T3+2T4≥0' and the related boundary-monotonicity claims are a missing verification, not a circular reduction: they do not assume the theorem and are checkable from the displayed definitions of T3,T4. Therefore there is no self-definitional, fitted-input, or self-citation circularity.

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

The paper introduces no new fitted parameters and no new postulated entities. It works within the standard framework of bi-univalent functions and subordination.

assumptions (5)

standard math Lemma 1.2 (Kanas et al.) parameterizes Schwarz function coefficients c2, c3 in terms of x, ξ with |x| ≤ 1, |ξ| ≤ 1.
Used to express c2, c3, d2, d3 in the proof of Theorem 2.1.
standard math Schwarz lemma: for analytic u(z) with u(0)=0 and |u(z)|<1, the coefficients satisfy |cn| ≤ 1, with equality only for rotations.
Provides the bounds |c1| ≤ 1 and the coefficient constraints used throughout the derivation.
standard math Subordination principle: f ≺ φ means f(z) = φ(u(z)) for a Schwarz function u, and coefficient matching through the expansion of φ(u(z)).
Justifies the coefficient equations (2.9)-(2.14) that the whole proof rests on.
standard math Inverse function coefficients: A2 = -a2, A3 = 2a2^2 - a3, A4 = -5a2^3 + 5a2 a3 - a4.
Used to set up the second subordinate condition for f^{-1}.
domain assumption Parameter domain assumptions: λ ≥ 1, 0 ≤ δ ≤ 1, τ ∈ C*, B1 > 0, B2, B3 real, φ(U) symmetric and starlike with φ(0)=1.
These define the class HΣ. The unproved inequalities T3+2T4 ≥ 0 and T3+T4 ≥ 0 depend on α ≤ β ≤ γ, which follows from λ ≥ 1 and δ ≥ 0.

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Pith. "Pith review of Upper bound of second Hankel determinant of subclass of bi-univalent functions defined by subordination." pith.science (2026). https://pith.science/paper/KP6SEKXV

@misc{pith2026190807350,
  author       = {Pith},
  title        = {Pith review of: Upper bound of second Hankel determinant of subclass of bi-univalent functions defined by subordination},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KP6SEKXV}},
  note         = {Machine review of arXiv:1908.07350}
}

abstract

In this paper, new class of bi-univalent functions are introduced. Upper bound of the second Hankel determinant $|H_2(2)|$ of subclass of bi-univalant functions class $\Sigma$, which defined by subordination, investigated. Furthermore, some results concluded as a special case of our main results and corrected some previous researchers results which investigated by miscalculation.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Pith tools

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

[1] [1]

Mat., 29(2018), 78, 1265-1271

Al-Hawary , T., Frasin, B.A., Yousef, F.: Coeﬃcients estimates for certain classes of analytic functions of complex order, Afr. Mat., 29(2018), 78, 1265-1271

work page 2018

[2] [2]

Amourah, A.A., Yousef, F., Al-Hawary , T., Darus, M.: A certain fractional derivative operator for p-valent functions and new class of analytic functions with neg ative coe ﬃcients, Far East Journal of Mathematical Sciences, 99(2016), 1, 75-87

work page 2016

[3] [3]

Amourah, A.A., Yousef, F., Al-Hawary , T., Darus, M.:On a class of p-valent non-Bazilevi˘ c functions of order µ+ iβ, International Journal of Mathematical Analysis, 10(2016), 15, 701-710

work page 2016

[4] [4]

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

work page 2017

[5] [5]

Egyptian Math

Bansal, D., Srivastava, H.M.: Coeﬃcient estimates for a subclass of analytic and bi-univalent functions, J. Egyptian Math. Soc., 23(2015), 2, 242-246

work page 2015

[6] [6]

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

work page 2016

[7] [7]

C ¸ aglar, M., Deniz, E., Srivastava, H.M.: Second Hankel determinant for certain subclasses of bi- univalent functions, Turk. J. Math., 41 (2017), 694-706

work page 2017

[8] [8]

Cantor, D.G.: Power series with integral coe ﬃcients, B. Am. Math. Soc., 69 (1963), 362-366

work page 1963