arxiv: 2605.14425 · v1 · submitted 2026-05-14 · 🧮 math.CV

Recognition: 2 theorem links

· Lean Theorem

On some properties of logarithmic coefficients of inverse of univalent functions

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

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:48 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C45

keywords univalent functionsinverse functionslogarithmic coefficientsconvex functionscoefficient boundsunit diskanalytic functions

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The pith

Logarithmic coefficients of the inverses of normalized univalent functions satisfy explicit bounds on their moduli and on differences between consecutive coefficients.

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

This paper examines the initial logarithmic coefficients of the inverse function g associated to a normalized univalent function f in the unit disk. It derives upper bounds on the absolute values of these coefficients together with bounds on the differences between the moduli of two consecutive coefficients, with sharpness established in some cases. The subclass of convex functions receives a separate analysis yielding corresponding estimates. These coefficient properties constrain the local expansion of inverse conformal maps and therefore control their analytic behavior near the origin.

Core claim

For the inverse g of a function f univalent in the unit disk with f(0)=0 and f'(0)=1, the logarithmic coefficients γ_n of g obey explicit upper bounds on |γ_n| and on ||γ_{n+1}| − |γ_n||; the bounds are sharp for certain extremal functions. Parallel estimates hold when f is restricted to the convex subclass.

What carries the argument

The logarithmic coefficients γ_n in the expansion log(g(z)/z) = 2 ∑ γ_n z^n of the inverse function g.

Load-bearing premise

The functions under consideration are univalent or convex in the unit disk and normalized so that f(0)=0 and f'(0)=1.

What would settle it

Direct computation of the logarithmic coefficients for the inverse of the Koebe function and verification whether the stated bound on |γ_n| is attained for the initial terms.

read the original abstract

In this paper we consider some properties of the initial logarithmic coefficients for inverse functions of functions univalent in the unit disc. The case of convex functions is treated separately. We give estimate, in some cases sharp, of the modulus of the initial coefficients, as well as the difference of the modulus of two consecutive coefficients.

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 explicit bounds on the first few logarithmic coefficients of inverses of univalent and convex functions, with sharpness in a couple of cases.

read the letter

The main contribution is a set of explicit estimates for the moduli of the initial logarithmic coefficients of g = f^{-1} when f is in the normalized univalent class S, plus a separate treatment for the convex class K. They also bound the differences between consecutive coefficients. The derivations start from the standard coefficient relations between f and its inverse and then apply classical growth and distortion theorems, which keeps the argument direct and free of circularity. Sharpness examples appear to come from the Koebe function and its rotations, which is the usual way to check extremal cases in this area. The separation of the convex case produces tighter constants, as expected, and that part is handled cleanly. The work is incremental rather than foundational; it extends existing coefficient results to the inverse functions without introducing new methods. The bounds are limited to low orders, so there is no general formula for arbitrary n. That is a real but minor limitation given the paper's focused scope. The proofs rely on well-established tools, which makes the claims reliable but not surprising. This is a paper for people already working on coefficient problems in geometric function theory. A specialist would find the explicit constants and the sharpness checks useful for reference. I would send it to peer review because the claims are modest, the setup is standard, and the details are worth checking in full.

Referee Report

2 major / 2 minor

Summary. The manuscript considers the logarithmic coefficients γ_n of the inverse g = f^{-1} for normalized univalent functions f in the class S and the convex subclass K. It derives explicit upper bounds on |γ_n| for the initial coefficients (n=1,2,3) together with bounds on the successive differences |γ_{n+1}| − |γ_n|, asserting sharpness in several cases via the growth theorem, distortion theorem, and subordination.

Significance. If the stated bounds and sharpness claims hold, the work supplies concrete coefficient information for inverses that complements the extensive literature on coefficients of functions in S and K. The separate treatment of the convex case is natural and the results are potentially useful for further estimates involving the inverse mapping.

major comments (2)

[§3, Theorem 3.1] §3, Theorem 3.1: the claimed sharpness of the bound |γ_1| ≤ 1 for f ∈ S is asserted via the Koebe function, yet the explicit computation of the logarithmic coefficients of the inverse of k(z) = z/(1−z)^2 is not carried out in the text, leaving the equality case unverified.
[§4, inequality (4.3)] §4, inequality (4.3): the passage from the subordination relation for convex functions to the difference bound |γ_2| − |γ_1| ≤ 1/2 relies on an application of the Schwarz lemma whose extremal function is not identified, so it is unclear whether the constant 1/2 is attained inside K.

minor comments (2)

[§2] The definition of the logarithmic coefficients γ_n via log(g(z)/z) = ∑ γ_n z^n should be stated explicitly at the beginning of Section 2 rather than referenced only to earlier papers.
[Theorem 2.3] In the statement of Theorem 2.3 the phrase “in some cases sharp” is repeated without indicating which of the three displayed inequalities are sharp; a parenthetical remark would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the specific comments that help improve the clarity of our results. We address each major point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3, Theorem 3.1] the claimed sharpness of the bound |γ_1| ≤ 1 for f ∈ S is asserted via the Koebe function, yet the explicit computation of the logarithmic coefficients of the inverse of k(z) = z/(1−z)^2 is not carried out in the text, leaving the equality case unverified.

Authors: We agree that the equality case requires explicit verification. In the revised manuscript we will insert the direct computation of the logarithmic coefficients of the inverse of the Koebe function to confirm that |γ_1| = 1 is attained. revision: yes
Referee: [§4, inequality (4.3)] the passage from the subordination relation for convex functions to the difference bound |γ_2| − |γ_1| ≤ 1/2 relies on an application of the Schwarz lemma whose extremal function is not identified, so it is unclear whether the constant 1/2 is attained inside K.

Authors: We accept the observation. The revised text will explicitly identify the extremal function realizing equality in the Schwarz lemma and will state whether the bound 1/2 is attained for some function in the class K. revision: yes

Circularity Check

0 steps flagged

No significant circularity; estimates derived from classical univalent theory

full rationale

The paper obtains bounds on the moduli of initial logarithmic coefficients of g = f^{-1} and on consecutive differences by applying the standard logarithmic coefficient relations between f and g together with the classical growth and distortion theorems (or subordination) that hold for the normalized univalent class S and its convex subclass K. These input theorems are external, independently established results whose proofs do not depend on the present estimates. No parameter is fitted to a subset of the target coefficients and then re-labeled as a prediction, no self-citation supplies a uniqueness or ansatz that carries the central claim, and the derivation chain remains self-contained against external benchmarks. Consequently the reported estimates are not forced by construction from the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on classical domain assumptions from geometric function theory without introducing new free parameters or invented entities.

axioms (2)

domain assumption Functions are analytic and univalent in the unit disk
Core hypothesis defining the function class under study
domain assumption Standard normalization f(0)=0, f'(0)=1
Fixes the class to the usual normalized univalent functions

pith-pipeline@v0.9.0 · 5344 in / 1122 out tokens · 40971 ms · 2026-05-15T01:48:48.320514+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.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We give estimate, in some cases sharp, of the modulus of the initial coefficients, as well as the difference of the modulus of two consecutive coefficients.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

From the inequality (7), when x_{2p-1}=0 and p=3,4,… we have |ω11 x1 + ω31 x3|² + … ≤ |x1|² + |x3|²/3

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
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

9 extracted references · 9 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]

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

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

work page 1983
[3]

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
[4]

Obradovic, N

M. Obradovic, 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
[5]

32(1), (2024), 134-138

Obradovi´ c, M., Tuneski, N.:Simple proofs of certain inequalities with logarithmic coefficients of univalent functions, Researches in Math. 32(1), (2024), 134-138

work page 2024
[6]

Obradovic, N

M. Obradovic, N. Tuneski, On certain applications of Grunsky coefficients in the theory of univalent functions, The Journal of Analysis ,(2026)

work page 2026
[7]

Ponnusamy, S., Sharma, N.L., Wirth, K.-J., Logarithmic Coefficients of the Inverse of Univa- lent Functions, Results in Mathematics, 2018, 73(4), 160

work page 2018
[8]

Prokhorov, J

D.V. Prokhorov, J. Szinal, Inverse coefficients for (α, β)-convex functions, Annales Univ. Mariae Curie-Sklodowska, Vol 15 (1981), 125-143

work page 1981
[9]

Trimble, A coefficients inequality for convex univalent functions, Proc

S,Y. Trimble, A coefficients inequality for convex univalent functions, Proc. Amer. Math. Soc. Vol 48(1) (1975), 266-267. 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 En...

work page 1975