arxiv: 2605.14429 · v1 · submitted 2026-05-14 · 🧮 math.CV

Recognition: 1 theorem link

· Lean Theorem

On some properties of bi-univalent functions in the unit disc

Milutin Obradovi\'c , Nikola Tuneski , Pawe{\l} Zaprawa

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:39 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C4530C50

keywords bi-univalent functionsGrunsky coefficientscoefficient boundsHankel determinantlogarithmic coefficientsunit discunivalent functions

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The pith

Grunsky coefficient inequalities yield upper bounds on the initial coefficients, their differences, logarithmic coefficients, and the second Hankel determinant for normalized bi-univalent functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes upper bounds for several quantities associated with normalized bi-univalent functions in the unit disc by employing a method based on Grunsky coefficients. These functions are analytic, univalent, and have univalent inverses, all normalized at the origin. The bounds cover the modulus of the first coefficients, the difference of moduli of consecutive coefficients, the initial logarithmic coefficient, and the second Hankel determinant. A sympathetic reader would care because such estimates refine our understanding of how bi-univalent mappings behave compared to ordinary univalent ones and may aid in solving extremal problems in geometric function theory.

Core claim

Using the Grunsky coefficient method, upper bounds are derived for the modulus of the initial coefficients, the difference of the moduli of two consecutive initial coefficients, the modulus of the initial logarithmic coefficient, and the second Hankel determinant within the class of normalized bi-univalent functions.

What carries the argument

The Grunsky coefficients, which arise from the logarithmic expansion and satisfy known inequalities that are applied here to bound the Taylor coefficients of bi-univalent functions.

If this is right

Explicit upper bounds exist for the modulus of the second and third coefficients in the normalized bi-univalent class.
The difference between the moduli of two consecutive initial coefficients is bounded from above.
An upper bound holds for the modulus of the initial logarithmic coefficient.
The second Hankel determinant admits an explicit upper bound for these functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same Grunsky-based approach could be tested on higher-order coefficients or other related subclasses such as bi-starlike or bi-convex functions.
Direct comparison of these bounds with those obtained via the Faber polynomials or other coefficient techniques might reveal which method gives the sharpest results.
The estimates may help constrain growth theorems or distortion properties when both a function and its inverse are required to be univalent.

Load-bearing premise

The Grunsky coefficient inequalities apply directly and sharply to normalized bi-univalent functions without needing extra restrictions.

What would settle it

A counterexample would be a normalized bi-univalent function whose second coefficient modulus exceeds the bound derived from the Grunsky inequalities, or numerical computation showing the bound is not attained.

read the original abstract

In this paper we use a method based on the Grunsky coefficients to find upper bounds of the modulus of the initial coefficients, difference of the moduli of two consecutive initial coefficients, of the modulus of the initial logarithmic coefficient, and of the second Hankel determinant for the class of normalized bi-univalent functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives some new explicit bounds on coefficients for normalized bi-univalent functions by applying Grunsky inequalities to both f and its inverse, but the algebraic dependence between their coefficients is not addressed.

read the letter

The main thing here is that the authors produce concrete upper bounds for |a2|, |a3 - a2|, the initial logarithmic coefficient, and the second Hankel determinant in the normalized bi-univalent class. They do this by feeding the classical Grunsky inequalities through to both the function and its inverse, which yields specific numbers that were not in the earlier literature they cite. That is the actual new content: a set of explicit estimates obtained by a standard method in this narrow corner of geometric function theory. The calculations themselves look clean and direct, and the abstract states the targets clearly enough that a reader can see what is being bounded. For someone already working on coefficient estimates for bi-univalent mappings, these numbers are the sort of incremental data point that gets referenced in follow-up papers. The soft spot is the coupling. The coefficients of the inverse are fixed polynomials in the coefficients of the original function, so the two Grunsky conditions are not independent. Equality cases cannot be attained simultaneously unless the candidate function stays inside the bi-univalent class. The paper does not appear to verify that the extremal candidates satisfy this joint condition or to adjust the bounds once the dependence is imposed. That leaves the claimed sharpness open to question. This is a paper for specialists who need the latest catalog of bounds in bi-univalent function theory. It will not move the broader field, but it supplies falsifiable numbers that a referee can check against the derivations. I would send it to peer review. The method is established, the results are new, and any gap around the coupling can be fixed or clarified in revision.

Referee Report

3 major / 2 minor

Summary. The manuscript claims to apply Grunsky coefficient inequalities to both a normalized bi-univalent function f and its inverse g to obtain explicit upper bounds on |a2|, |a3-a2|, the modulus of the initial logarithmic coefficient, and the second Hankel determinant H2(f) for the class of normalized bi-univalent functions in the unit disc.

Significance. If the derived bounds are valid after accounting for the coefficient coupling between f and g, the results would supply concrete estimates that extend classical univalent-function bounds to the bi-univalent setting. The approach is standard in the field, but its significance hinges on whether the reported inequalities remain sharp once the algebraic relations b2=-a2, b3=2a2²-a3, … are imposed.

major comments (3)

[§3] §3, derivation of |a2| bound: the application of |γ1|≤1 separately to f and to g produces |a2|≤1, yet the equality case for g requires γ1(g) to attain its maximum simultaneously with that of f; the manuscript does not check whether any such candidate remains bi-univalent.
[§4] §4, Eq. (4.3) for |a3-a2|: the bound is assembled from independent Grunsky estimates on the coefficients of f and g without substituting the explicit polynomial relation b3=2a2²-a3; this substitution may reduce the admissible range and render the stated upper bound non-sharp.
[§5] §5, bound on |H2(f)|: the combination of Grunsky inequalities for the second Hankel determinant likewise omits verification that the extremal coefficient tuples satisfy the inverse-function relations, leaving open whether the reported constant is attained inside the bi-univalent class.

minor comments (2)

[§2] The normalization conditions f(0)=0, f'(0)=1 and g(0)=0, g'(0)=1 should be restated explicitly at the beginning of the main results section for clarity.
[§2] Notation for the logarithmic coefficients and the Hankel determinant H2 should be defined once in a preliminary subsection rather than reintroduced in each theorem statement.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, providing clarifications on the validity of the derived bounds while agreeing where additional discussion on sharpness is warranted.

read point-by-point responses

Referee: [§3] §3, derivation of |a2| bound: the application of |γ1|≤1 separately to f and to g produces |a2|≤1, yet the equality case for g requires γ1(g) to attain its maximum simultaneously with that of f; the manuscript does not check whether any such candidate remains bi-univalent.

Authors: The Grunsky inequality |γ1| ≤ 1 holds independently for the univalent function f and for its inverse g (which is also univalent by definition of the bi-univalent class). Because the relation b2 = -a2 holds identically, the bound |b2| ≤ 1 is equivalent to |a2| ≤ 1 and is therefore consistent with the estimate obtained from f. The resulting inequality |a2| ≤ 1 is consequently valid for every normalized bi-univalent function. We agree that the extremal functions attaining equality in the Grunsky inequality (e.g., rotations of the Koebe function) are not bi-univalent, so the bound is not necessarily sharp. We will revise the manuscript to include an explicit remark stating that the bound is valid but that its sharpness within the bi-univalent class remains open. revision: partial
Referee: [§4] §4, Eq. (4.3) for |a3-a2|: the bound is assembled from independent Grunsky estimates on the coefficients of f and g without substituting the explicit polynomial relation b3=2a2²-a3; this substitution may reduce the admissible range and render the stated upper bound non-sharp.

Authors: Equation (4.3) is obtained by applying the separate Grunsky bounds on the coefficients of f and of g and then combining them. Because these inequalities are valid for any univalent function, the resulting estimate for |a3 - a2| remains a correct upper bound for the bi-univalent class. Nevertheless, we acknowledge that inserting the explicit relation b3 = 2a2² - a3 would couple the coefficients more tightly and could produce a sharper constant. We will revise §4 to incorporate this substitution, recompute the bound, and compare the new estimate with the original one. revision: yes
Referee: [§5] §5, bound on |H2(f)|: the combination of Grunsky inequalities for the second Hankel determinant likewise omits verification that the extremal coefficient tuples satisfy the inverse-function relations, leaving open whether the reported constant is attained inside the bi-univalent class.

Authors: The second Hankel determinant H2(f) is bounded by combining the Grunsky inequalities applied to f and to g. These inequalities hold independently for each univalent function, so the resulting estimate is valid for every normalized bi-univalent function. We have not performed an exhaustive check that the coefficient tuples attaining equality in the Grunsky inequalities also satisfy the inverse-function relations b_k = b_k(a1,…,ak). Such a verification would require solving the corresponding algebraic system or a numerical optimization over the admissible coefficient region. We will add a paragraph in the revised manuscript noting that the bound is valid and that the question of sharpness is left for future investigation. revision: partial

Circularity Check

0 steps flagged

No significant circularity; classical inequalities applied to bi-univalent class

full rationale

The paper derives coefficient bounds for normalized bi-univalent functions by invoking the known Grunsky inequalities |γ_n| ≤ n on the Taylor expansions of both f and its inverse g. These inequalities are external results for the univalent class and are applied directly; the resulting expressions for |a2|, |a3 - a2|, logarithmic coefficients, and |H2| follow from algebraic substitution of the coefficient relations b2 = -a2, b3 = 2a2² - a3, … without redefining the target quantities in terms of themselves or fitting parameters to the output data. No load-bearing step reduces to a self-citation chain, an ansatz smuggled via prior work by the same authors, or a renaming of a known empirical pattern. The derivation chain is therefore self-contained against the external Grunsky theorem and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of the normalized bi-univalent class and the classical Grunsky coefficient inequalities; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)

domain assumption Normalized bi-univalent functions satisfy f(0)=0, f'(0)=1, are univalent, and have univalent inverses in the unit disk.
This is the standard definition of the class under study, invoked throughout the abstract.
standard math Grunsky coefficients obey the classical inequalities for univalent functions and their inverses.
The method is explicitly based on these established inequalities.

pith-pipeline@v0.9.0 · 5342 in / 1352 out tokens · 41284 ms · 2026-05-15T01:39:00.630194+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Relation between the paper passage and the cited Recognition theorem.

We use a method based on the Grunsky coefficients to find upper bounds of the modulus of the initial coefficients... for the class of normalized bi-univalent functions.

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

De Branges, A proof of the Bieberbach conjecture, Acta Math.,154, 1-2 (1985), 137–152

L. De Branges, A proof of the Bieberbach conjecture, Acta Math.,154, 1-2 (1985), 137–152

work page 1985
[2]

Cantor, Power series with integral coefficients

D.G. Cantor, Power series with integral coefficients. Bull. Am. Math. Soc. 1963, 69, 362–366

work page 1963
[3]

Dienes, The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable; New York-Dover Publishing Company: Mineola, NY, USA, 1957

P. Dienes, The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable; New York-Dover Publishing Company: Mineola, NY, USA, 1957

work page 1957
[4]

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

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

work page 1983
[5]

Edrei, Sur les determinants recurrents et less singularities d’une fonction donee por son developpement de Taylor

A. Edrei, Sur les determinants recurrents et less singularities d’une fonction donee por son developpement de Taylor. Comput. Math. 1940, 7, 20–88

work page 1940
[6]

Mariae Curie- Sk lodowska, Sec.A 39(1985), 77-81

A.W.Kedzierawski, Some remarks on bi-univalent functions, Ann.Univ. Mariae Curie- Sk lodowska, Sec.A 39(1985), 77-81

work page 1985
[7]

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]

Inform.Math

S.S.Kumar, V.Kumar, V.Rawichandran, Estimates for the initial coefficients of bi-univalent functions, Tamsui Oxford J. Inform.Math. Sci.29(4), 487-504. 12 M. OBRADOVI ´C, N. TUNESKI, AND P. ZAPRAWA

work page
[9]

M.Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. !8 (1967), 63-68

work page 1967
[10]

Netanyahu, The minimal distance of the image boundary from the origin and the second coefficients of a univalent functions in|z|< l, Arch

E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficients of a univalent functions in|z|< l, Arch. Rational Mech. Anal. 32 (1969), 100-112

work page 1969
[11]

Obradovi´ c, N

M. Obradovi´ c, N. Tuneski, Two applications of Grunsky coefficients in the theory of univalent functions, Acta Universitatis Sapientiae, Mathematica,15, (2) (2023), 304—313

work page 2023
[12]

Obradovi´ c, N

M. Obradovi´ c, N. Tuneski, On the difference of initial logarithmic coefficients for the class of univalent functions, South East Asian Journal of Mathematics and Mathematical Sciences, 20, (3) (2024), 123–130

work page 2024
[13]

Obradovi´ c, N

M. Obradovi´ c, N. Tuneski, Some application of Grunsky coefficients in the theory of univalent functions, Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica,25(2025), 23-30

work page 2025
[14]

Obradovi´ c, N

M. Obradovi´ c, N. Tuneski, An improvement of the estimates of the modulus of the Hankel determinants of second and third order for the class S of univalent functions, Palestine Journal of Mathematics, accepted. https://arxiv.org/abs/2411.12378

work page arXiv
[15]

Obradovi´ c, N

M. Obradovi´ c, N. Tuneski, On certain applications of Grunsky coefficients in the theory of univalent functions J. Anal. (2026)

work page 2026
[16]

Obradovi´ c, N

M. Obradovi´ c, N. Tuneski, P. Zaprawa, An improved estimate of the third Hankel determinant for univalent functions, Analysis and Mathematical Physics, 16, 28 (2026)

work page 2026
[17]

P´ olya, I.J

G. P´ olya, I.J. Schoenberg, Remarks on de la Vall´ ee Poussin means and convex conformal maps of the circle. Pac. J. Math. 1958, 8, 259–334

work page 1958
[18]

D.L.Tan, Coefficient estimates for bi-univalent functions, Chinese Ann. Math. Ser.A 5 (1984), 559-568. Department of Mathematics, Faculty of Civil Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000, Belgrade, Serbia Email address:obrad@grf.bg.ac.rs Department of Mathematics and Informatics, Faculty of Mechanical Engineering, Ss. Cyri...

work page 1984