The solution of the Brannan conjecture

Erhan Deniz; Murat \c{C}aglar; R\'obert Sz\'asz

arxiv: 1906.09498 · v1 · pith:HP65IUTFnew · submitted 2019-06-22 · 🧮 math.CV

The solution of the Brannan conjecture

Erhan Deniz , Murat \c{C}aglar , R\'obert Sz\'asz This is my paper

Pith reviewed 2026-05-25 17:37 UTC · model grok-4.3

classification 🧮 math.CV

keywords Brannan conjectureMaclaurin seriesintegral estimatesanalytic functionscoefficient boundscomplex analysis

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The pith

Brannan's conjecture is proved using a Maclaurin series expansion and an integral estimate.

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

The paper supplies the final step toward proving Brannan's conjecture. It expands the relevant functions in Maclaurin series and bounds a chosen integral to reach the required inequality. A sympathetic reader would care because the conjecture addresses coefficient relations for analytic functions inside the unit disk. If the argument holds, it settles a long-standing question on the extremal behavior of those coefficients.

Core claim

Using a Mac-Laurin development of the function together with an adequate estimation of an integral, the authors complete the proof of Brannan's conjecture and thereby establish its full statement for the functions under consideration.

What carries the argument

Mac-Laurin series development combined with estimation of a suitable integral, used to derive the coefficient inequality asserted by the conjecture.

If this is right

The coefficient bounds stated in Brannan's conjecture hold for all functions in the class.
The same expansion-plus-estimate technique confirms the relations among Taylor coefficients for the functions considered.
Related coefficient problems in the same function class become accessible by analogous integral bounds.

Where Pith is reading between the lines

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

Direct numerical checks on standard examples such as the Koebe function could serve as quick consistency tests for the derived bounds.
The method might extend without major change to coefficient problems for nearby classes of analytic functions.
If the bound is sharp, the proof would identify the functions that attain equality.

Load-bearing premise

The Mac-Laurin development combined with the chosen integral estimate is sufficient to establish the full statement of the Brannan conjecture without additional restrictions or unstated conditions on the functions involved.

What would settle it

An explicit analytic function satisfying the conjecture's hypotheses but violating the predicted coefficient relation would show the argument does not hold.

read the original abstract

We make the final step to give a proof for the Brannan's conjecture. The basic tool of the study is a Mac-Laurin development and an adequately estimation of an integral.

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 claims to complete the proof of Brannan's conjecture with a Mac-Laurin expansion plus an integral estimate, but the abstract supplies no explicit bound, remainder, or check against extremals.

read the letter

The main point is that the authors say they have supplied the last piece needed to prove Brannan's conjecture. They rely on a Mac-Laurin development followed by an estimate of an integral. If that estimate turns out to be both sharp and uniform over the whole class, the claim would close a named problem in geometric function theory after earlier partial results. That is the only concrete contribution visible here. The approach itself is standard for coefficient-type questions in univalent functions, so the authors are using familiar tools rather than introducing new machinery. The description is brief, which makes it hard to judge whether the work actually succeeds. No truncation order is stated, no remainder term appears, and there is no indication that the bound was compared with the Koebe function or other known extremals. The stress-test note correctly flags the risk that an unspecified integral estimate may fail to control every case the conjecture requires. Without those checks the argument could contain gaps that only surface on close reading. The paper is written for people already working on Brannan's conjecture and related coefficient problems. A reader who knows the prior literature could see whether the final step is handled correctly, but only if the full derivation is supplied. I would send the manuscript to referees. The conjecture is old enough that a claimed resolution merits expert scrutiny even if the current write-up needs expansion and verification steps.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to complete the proof of Brannan's conjecture in geometric function theory by means of a Maclaurin series development together with an estimate for a certain integral.

Significance. A correct resolution of the Brannan conjecture would constitute a substantial advance in complex analysis. The approach via series expansion and integral estimation is in principle capable of yielding such a result, but the manuscript supplies neither the explicit expansion, the remainder term, nor any uniformity or sharpness verification, so the significance cannot be assessed from the given text.

major comments (2)

The abstract states that the proof rests on a Mac-Laurin development and 'an adequately estimation of an integral,' yet supplies neither the order of truncation, the explicit form of the integral, nor the error bound. Without these, it is impossible to check whether the estimate is uniform over the full class of functions and closes the conjecture for every coefficient or mean addressed.
No comparison with known extremal functions (e.g., the Koebe function or other candidates for equality cases) is mentioned, leaving open the possibility that the chosen integral bound fails to be sharp or requires unstated restrictions on the functions.

minor comments (1)

The phrasing 'Mac-Laurin development' and 'adequately estimation' should be corrected to standard mathematical English ('Maclaurin expansion' and 'adequate estimate').

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed comments on our manuscript. We provide point-by-point responses below and indicate where revisions will be made to address the concerns.

read point-by-point responses

Referee: The abstract states that the proof rests on a Mac-Laurin development and 'an adequately estimation of an integral,' yet supplies neither the order of truncation, the explicit form of the integral, nor the error bound. Without these, it is impossible to check whether the estimate is uniform over the full class of functions and closes the conjecture for every coefficient or mean addressed.

Authors: We agree that the abstract is insufficiently detailed regarding the truncation order, the explicit integral, and the error bound. The manuscript body contains the Maclaurin development and integral estimation intended to prove the conjecture, but we acknowledge that explicit statements of these elements and verification of uniformity are necessary for the reader to verify the result. We will revise the manuscript to include the explicit expansion, the form of the integral, the remainder estimate, and a statement on uniformity over the class of functions. revision: yes
Referee: No comparison with known extremal functions (e.g., the Koebe function or other candidates for equality cases) is mentioned, leaving open the possibility that the chosen integral bound fails to be sharp or requires unstated restrictions on the functions.

Authors: The manuscript does not include an explicit comparison with extremal functions such as the Koebe function. We will add a remark or subsection discussing the sharpness of the integral bound by comparing with known extremal cases to confirm that the estimate holds with equality in the appropriate cases. revision: yes

Circularity Check

0 steps flagged

No circularity: proof claim rests on Mac-Laurin expansion plus integral estimate without reduction to fitted inputs or self-citation chains.

full rationale

The abstract and description present the argument as a direct analytic proof of the Brannan conjecture via Mac-Laurin series development combined with an integral estimate. No equations, parameter fits, or citations are supplied that would allow any step to reduce by construction to its own inputs. The derivation is therefore treated as self-contained pending external verification of the estimate's uniformity; no load-bearing self-citation or definitional loop is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the argument rests on the validity of the MacLaurin expansion for the functions under study and on the claim that a particular integral admits an adequate estimate; both are standard in complex analysis and no new entities or fitted parameters are mentioned.

axioms (2)

standard math MacLaurin series expansion applies to the functions considered in the conjecture
Standard tool for analytic functions in the unit disk.
domain assumption An adequate estimate of the relevant integral exists and suffices for the proof
The abstract identifies this estimation as the basic tool.

pith-pipeline@v0.9.0 · 5541 in / 1273 out tokens · 34227 ms · 2026-05-25T17:37:43.195673+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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Aharonov and S

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R. W. Barnard, U. C. Jayatilake, and A. Yu. Solynin, Brannan’s co n- jecture and trigonometric sums, Proc. Amer. Math. Soc. 143(5) (2015), 2117-2128

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R. W. Barnard, K. Pearce, and W. Wheeler, On a coeﬃcient conje cture of Brannan, Complex Var. Theory Appl. 33(1-4) (1997), 51-61

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D. A. Brannan, On coeﬃcient problems for certain power series, Pro- ceedings of the Symposium on Complex Analysis (Univ. Kent, Canter- bury, 1973), Cambridge Univ. Press, London, 1974, pp. 17-27. L ondon Math. Soc. Lecture Note Ser., No. 12

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[6]

U. C. Jayatilake, Brannans conjecture for initial coeﬃcients. C omplex Var. Elliptic Equ. 58(5)(2013), 685-694

work page 2013
[7]

Milcetich, On a coeﬃcient conjecture of Brannan, J

J.n G. Milcetich, On a coeﬃcient conjecture of Brannan, J. Math. Anal. Appl. 139(2)(1989), 515-522

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[8]

Ruscheweyh and L

S. Ruscheweyh and L. Salinas, On Brannans coeﬃcient conjectu re and applications, Glasg. Math. J. 49 (1)(2007), 45-52

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[9]

Sz´ asz R´ obert, On the Brannan’s conjecture, Mediterranean Journal of Mathematics, Accepted Paper. Murat C ¸ a˘ glar and Erhan Deniz Department of Mathematics Faculty of Science and Letters Kafkas University Kars Turkey 6 e-mail: mcaglar25@gmail.com e-mail : edeniz36@gmail.com R´ obert Sz´ asz Department of Mathematics and Informatics Sapientia Hungaria...

work page

[1] [1]

Aharonov and S

D. Aharonov and S. Friedland, On an inequality connected with the coeﬃcient conjecture for functions of bounded boundary rotat ion, Ann. Acad. Sci. Fenn. Ser. A I 524 (1972), 14p

work page 1972

[2] [2]

R. W. Barnard, Brannans coeﬃcient conjecture for certain po wer se- ries, Open problems and conjectures in complex analysis, Computat ional Methods and Function Theory (Valparaso, 1989), 1-26. Lecture notes in Math. 1435, Springer, Berlin, 1990

work page 1989

[3] [3]

R. W. Barnard, U. C. Jayatilake, and A. Yu. Solynin, Brannan’s co n- jecture and trigonometric sums, Proc. Amer. Math. Soc. 143(5) (2015), 2117-2128

work page 2015

[4] [4]

R. W. Barnard, K. Pearce, and W. Wheeler, On a coeﬃcient conje cture of Brannan, Complex Var. Theory Appl. 33(1-4) (1997), 51-61

work page 1997

[5] [5]

D. A. Brannan, On coeﬃcient problems for certain power series, Pro- ceedings of the Symposium on Complex Analysis (Univ. Kent, Canter- bury, 1973), Cambridge Univ. Press, London, 1974, pp. 17-27. L ondon Math. Soc. Lecture Note Ser., No. 12

work page 1973

[6] [6]

U. C. Jayatilake, Brannans conjecture for initial coeﬃcients. C omplex Var. Elliptic Equ. 58(5)(2013), 685-694

work page 2013

[7] [7]

Milcetich, On a coeﬃcient conjecture of Brannan, J

J.n G. Milcetich, On a coeﬃcient conjecture of Brannan, J. Math. Anal. Appl. 139(2)(1989), 515-522

work page 1989

[8] [8]

Ruscheweyh and L

S. Ruscheweyh and L. Salinas, On Brannans coeﬃcient conjectu re and applications, Glasg. Math. J. 49 (1)(2007), 45-52

work page 2007

[9] [9]

Sz´ asz R´ obert, On the Brannan’s conjecture, Mediterranean Journal of Mathematics, Accepted Paper. Murat C ¸ a˘ glar and Erhan Deniz Department of Mathematics Faculty of Science and Letters Kafkas University Kars Turkey 6 e-mail: mcaglar25@gmail.com e-mail : edeniz36@gmail.com R´ obert Sz´ asz Department of Mathematics and Informatics Sapientia Hungaria...

work page