Sharp coefficient Estimates for the class $\mathcal{S}_{\mathcal{AP}}^{*}$

Nabadwip Sarkar; Pradip Das

arxiv: 2606.15488 · v2 · pith:H3EYOD2Lnew · submitted 2026-06-13 · 🧮 math.CV

Sharp coefficient Estimates for the class mathcal{S}_{mathcal{AP}}^(*)

Pradip Das , Nabadwip Sarkar This is my paper

Pith reviewed 2026-06-27 03:37 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C45

keywords starlike functionsinverse logarithmic coefficientsHankel determinantFekete-Szegő functionalHermitian-Toeplitz determinantsubordinationcoefficient estimatesMa-Minda class

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The pith

The paper derives sharp bounds for the initial inverse logarithmic coefficients Γ1, Γ2, Γ3 and related determinants for the starlike class S_AP^* defined by subordination to the apple-like function e^z √(1+z).

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

This paper investigates coefficient problems for the Ma-Minda starlike subclass S_AP^* defined by the subordination relation involving the apple-like function ψ_AP(z) = e^z √(1+z). Sharp bounds are established for the inverse logarithmic coefficients Γ1, Γ2, Γ3 and the difference |Γ2| - |Γ1|. The second-order inverse logarithmic Hankel determinant, the generalized Fekete-Szegő functional for all real parameters, and the third-order Hermitian-Toeplitz determinant are evaluated, with explicit extremal functions provided for each. These findings offer precise control over the coefficient behavior in this specific class of analytic functions.

Core claim

Sharp bounds are derived for the initial inverse logarithmic coefficients Γ1, Γ2, Γ3, and the successive modulus difference |Γ2|−|Γ1|. In addition, the second-order inverse logarithmic Hankel determinant, the generalized Fekete-Szegő functional over all real parameter domains, and the third-order Hermitian-Toeplitz determinant are evaluated. The corresponding extremal functions are explicitly determined for each functional.

What carries the argument

The apple-like subordination function ψ_AP(z)=e^z √(1+z) that defines membership in the class S_AP^* of starlike functions.

Load-bearing premise

The functions under consideration must satisfy the subordination condition that places them in the class S_AP^* defined by the apple-like function.

What would settle it

Finding a function in S_AP^* for which the inverse logarithmic coefficient Γ3 exceeds the sharp bound claimed in the paper would disprove the result.

Figures

Figures reproduced from arXiv: 2606.15488 by Nabadwip Sarkar, Pradip Das.

**Figure 1.** Figure 1: Conformal mapping of the open unit disk D onto the symmetric apple-like Ma–Minda target domain under the generating boundary function ψAP(z) = e z √ 1 + z. The primary objective of this paper is to investigate several extremal problems within the class S ∗ AP. Specifically, we study its inverse logarithmic coefficients, consecutive coefficient differences, and structural determinants to map the geometric b… view at source ↗

**Figure 2.** Figure 2: Conformal mapping profile from the open unit disk D to the starlike target domain f0(D) associated with the sharp boundary constants. 4. Bounds for the differences of logarithmic inverse coefficients for S ∗ AP The Bieberbach conjecture, proved by de Branges [10], states that the Taylor coefficients of any function f ∈ S of the form (1.1) satisfy the bound |an| ≤ n for all n ≥ 2, with equality holding onl… view at source ↗

**Figure 3.** Figure 3: Conformal mapping from the unit disk D onto the two-fold symmetric extremal domain f1(D). To establish the sharpness of the lower bound in (4.1), we consider the function f2 ∈ S∗ AP defined by f2(z) = z exp Z z 0 e ω(t)p 1 + ω(t) − 1 t dt! , (4.4) where ω(z) = z z + 2 √ √ 3 41 / 1 + 2 √ √ 3 41 z . The coefficients of the corresponding function p(z) = 1+ω(z) 1−ω(z) are c1 = 4 √ √ 3 41 and c2 = 2. Th… view at source ↗

**Figure 4.** Figure 4: Conformal mapping from the unit disk D onto the asymmetric extremal domain f2(D). 5. Hankel determinants for the inverse logarithmic coefficients for S ∗ AP Theorem 5.1. Let f ∈ S∗ AP be given by (1.1). Then [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

read the original abstract

We investigate several classic coefficient problems for the Ma--Minda starlike subclass $\mathcal{S}_{\mathcal{AP}}^{*}$ defined by the apple-like subordination function $\psi_{\mathcal{AP}}(z)=e^{z}\sqrt{1+z}$. Sharp bounds are derived for the initial inverse logarithmic coefficients $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, and the successive modulus difference $|\Gamma_2|-|\Gamma_1|$. In addition, we evaluate the second-order inverse logarithmic Hankel determinant, the generalized Fekete--Szeg\"o functional over all real parameter domains, and the third-order Hermitian--Toeplitz determinant. The corresponding extremal functions are explicitly determined for each functional.

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

Routine sharp bounds on inverse log coefficients for one narrow Ma-Minda class, done with standard methods and explicit extremals.

read the letter

The paper works out sharp bounds for the first three inverse logarithmic coefficients, their modulus difference, the second-order inverse log Hankel determinant, the generalized Fekete-Szegő functional over real parameters, and the third-order Hermitian-Toeplitz determinant, all inside the starlike class defined by subordination to ψ_AP(z) = e^z √(1+z).

The authors follow the usual route: expand the composition with a Schwarz function, extract coefficients, and maximize over the admissible region. They also name the extremal functions for each functional. That part is clean and complete on the terms they set.

What is new is simply the application to this particular ψ_AP, which had not received these calculations before. The methods themselves are textbook for the area.

The limitation is the narrow scope. The class is one more specific choice among many possible subordination functions, and nothing in the results points to a general pattern or a new technique that would apply elsewhere. The apple-like function does not appear to connect to other problems in a way that would make these bounds useful outside the immediate literature on Ma-Minda subclasses.

The derivations look internally consistent from the description, with no circular steps or missing normalizations. The explicit extremals provide independent grounding if the algebra holds.

This is the kind of paper that interests only people already tracking coefficient problems for every new Ma-Minda class. A specialist might want the numbers for comparison, but it does not reorganize anything larger.

It is solid enough on its own terms to go to a referee who knows the standard toolkit. The claims are concrete and checkable.

Referee Report

0 major / 2 minor

Summary. The manuscript investigates several classic coefficient problems for the Ma-Minda starlike subclass S_AP^* defined by the apple-like subordination function ψ_AP(z)=e^z √(1+z). Sharp bounds are derived for the initial inverse logarithmic coefficients Γ1, Γ2, Γ3, and the successive modulus difference |Γ2|−|Γ1|. In addition, the second-order inverse logarithmic Hankel determinant, the generalized Fekete-Szegő functional over all real parameter domains, and the third-order Hermitian-Toeplitz determinant are evaluated, with corresponding extremal functions explicitly determined for each functional.

Significance. If the derivations hold, the results add explicit sharp estimates and extremal functions to the literature on coefficient problems for specific subordination-defined starlike classes. The explicit identification of extremal functions for each functional is a strength that supports verifiability of the bounds.

minor comments (2)

[Introduction] The introduction would benefit from a short paragraph situating ψ_AP(z) relative to other recently studied apple-like or exponential-type subordinants (e.g., those in the references on Ma-Minda classes).
[Section 3] In the coefficient expansions leading to the inverse logarithmic coefficients, the recurrence relations for the coefficients of log(f(z)/z) should be stated explicitly before the maximization arguments begin.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. No specific major comments were provided in the report, so we have no individual points to address at this time. We are prepared to incorporate any minor editorial suggestions that may arise during the revision process.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds from the standard subordination definition of the class S_AP^* via the given apple-like function, followed by series expansion of the composition with a Schwarz function and maximization over the coefficient region of the Schwarz function. Sharpness is asserted via explicit identification of extremal functions, which constitutes independent grounding rather than a reduction to the input by construction. No self-definitional steps, fitted parameters renamed as predictions, load-bearing self-citations, or imported uniqueness theorems appear in the chain. The central claims remain self-contained against external benchmarks of univalent function theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the analyticity of the functions, the definition of the Ma-Minda starlike class via subordination to ψ_AP, and standard lemmas on coefficient bounds that are assumed from prior literature.

axioms (2)

standard math Functions are analytic and normalized in the unit disk with f(0)=0, f'(0)=1.
Implicit in the definition of the class S_AP^* and all coefficient problems in the abstract.
domain assumption Subordination f'(z)/f(z) ≺ ψ_AP(z) defines the class.
Stated directly in the abstract as the defining property of S_AP^*.

pith-pipeline@v0.9.1-grok · 5648 in / 1350 out tokens · 32015 ms · 2026-06-27T03:37:24.878743+00:00 · methodology

discussion (0)

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Works this paper leans on

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B., & Mandal, S

Ahamed, M. B., & Mandal, S. (2026a). Sharp estimates of logarithmic coefficients for a certain class of starlike functions.arXiv preprint arXiv:2603.15659

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K.: Successive coefficients for spirallike and related functions

Arora, V., Ponnusamy, S., & Sahoo, S. K.: Successive coefficients for spirallike and related functions. Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. Ser. A Mat.113(4), 2969–2979 (2019)

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K., & Sugawa, T.: Successive coefficients for functions in the spirallike family.Bull

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2023

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Bulboaca, T., Obradovic, M., & Tuneski, N.: Simple proofs of certain results on generalized Fekete–Szeg˝ o functional in the classS.Anal. Math. Phys.15, 102 (2025)

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E., Kowalczyk, B., & Lecko, A.: Sharp bounds of some coefficient functionals over the class of functions convex in the direction of the imaginary axis.Bull

Cho, N. E., Kowalczyk, B., & Lecko, A.: Sharp bounds of some coefficient functionals over the class of functions convex in the direction of the imaginary axis.Bull. Aust. Math. Soc.100, 86–96 (2019)

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E., Kumar, S., & Kumar, V.: Hermitian-Toeplitz and Hankel determinants for certain starlike functions.Asian-European J

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2022

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Choi, J. H., Kim, Y. C., & Sugawa, T.: A general approach to the Fekete–Szeg˝ o problem.J. Math. Soc. Japan59, 707–727 (2007)

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S., Lecko, A., Sim, Y

Cudna, K., Kwon, O. S., Lecko, A., Sim, Y. J., & Smiarowska, B.: The second and third-order Hermitian Toeplitz determinants for starlike and convex functions of orderβ.Bol. Soc. Mat. Mex. (3)26(2), 361–375 (2020)

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S., Lecko, A., C ¸ eki¸ c, B., & S ¸eker, B.: The second Hankel determinant of logarithmic coefficients for strongly Ozaki close-to-convex functions.Bull

Eker, S. S., Lecko, A., C ¸ eki¸ c, B., & S ¸eker, B.: The second Hankel determinant of logarithmic coefficients for strongly Ozaki close-to-convex functions.Bull. Malays. Math. Sci. Soc.46, Art. 183 (2023)

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S., Lecko, A., Sim, Y

Kowalczyk, B., Kwon, O. S., Lecko, A., Sim, Y. J., & Smiarowska, B.: The third-order Hermitian Toeplitz determinant for classes of functions convex in one direction.Bull. Malays. Math. Sci. Soc.43(4), 3143–3158 (2020)

2020

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Kowalczyk, B., & Lecko, A.: Second Hankel determinant of logarithmic coefficients of convex and starlike functions.Bull. Aust. Math. Soc.105(3), 458–467 (2022)

2022

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Kowalczyk, B., & Lecko, A.: Second Hankel determinant of logarithmic coefficients of convex and starlike functions of orderα.Bull. Malays. Math. Sci. Soc.45, 727–740 (2022)

2022

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Kowalczyk, B., Lecko, A., & Smiarowska, B.: Sharp inequalities for Hermitian Toeplitz determinants for strongly starlike and strongly convex functions.J. Math. Inequal.15(1), 323–332 (2021)

2021

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ID 1154705 (2016)

Kucerovsky, D., Mousavand, K., & Sarraf, A.: On some properties of Toeplitz matrices.Cogent Math.3, Art. ID 1154705 (2016)

2016

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S., & Verma, N.: On estimation of Hankel determinants for certain class of starlike functions

Kumar, S. S., & Verma, N.: On estimation of Hankel determinants for certain class of starlike functions. Filomat38(10), 3469–3478 (2024)

2024

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Kumar, V.: Hermitian-Toeplitz determinants for certain classes of close-to-convex functions.Bull. Iran. Math. Soc.48, 1139–1150 (2022)

2022

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E.: Moduli difference of successive inverse and logarithmic coefficients for a class of close-to-convex functions.Asian-Eur

Kumar, V., & Cho, N. E.: Moduli difference of successive inverse and logarithmic coefficients for a class of close-to-convex functions.Asian-Eur. J. Math.16(2023)

2023

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E.: Hermitian-Toeplitz determinants for functions with bounded turning.Turkish J

Kumar, V., & Cho, N. E.: Hermitian-Toeplitz determinants for functions with bounded turning.Turkish J. Math.45(6), 2678–2687 (2021)

2021

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Kumar, V., & Kumar, S.: Bounds on Hermitian-Toeplitz and Hankel determinants for strongly starlike functions.Bol. Soc. Mat. Mex. (3)27(2), Art. 55, 14 pp. (2021)

2021

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E.: Sharp estimation of Hermitian-Toeplitz determinants for Janowski type starlike and convex functions.Miskolc Math

Kumar, V., Srivastava, R., & Cho, N. E.: Sharp estimation of Hermitian-Toeplitz determinants for Janowski type starlike and convex functions.Miskolc Math. Notes21(2), 939–952 (2020)

2020

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2024

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Lecko, A., & Partyka, D.: A generalized Fekete–Szeg˝ o functional and initial successive coefficients of univalent functions.Bull. Sci. Math.197, 103527 (2024)

2024

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Lecko, A., & ´Smiarowska, B.: Second Hankel determinant of logarithmic coefficients of inverse strongly starlike functions.Proc. Edinb. Math. Soc.67, 1196–1211 (2024)

2024

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J.: Coefficient problems in the subclasses of close-to-star functions.Results Math

Lecko, A., & Sim, Y. J.: Coefficient problems in the subclasses of close-to-star functions.Results Math. 74, 104 (2019). SHARP COEFFICIENT ESTIMATES FOR THE CLASSS ∗ AP 21

2019

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J., & Zlotkiewicz, E

Libera, R. J., & Zlotkiewicz, E. J.: Coefficient bounds for the inverse of a function with derivative inP. Proc. Amer. Math. Soc.87(2), 251–257 (1983)

1983

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Leung, Y.: Successive coefficients of starlike functions.Bull. Lond. Math. Soc.10, 193–196 (1978)

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Methods Funct

Li, M., & Sugawa, T.: A note on successive coefficients of convex functions.Comput. Methods Funct. Theory17(2), 179–193 (2017)

2017

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