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arxiv: 2606.10186 · v3 · pith:73RKKRZVnew · submitted 2026-06-08 · 🧮 math.CV

Sharp Coefficient Estimates for the Exponential Starlike class mathcal{S}_(ex)^(ast)

Pradip Das , Nabadwip Sarkar This is my paper

Pith reviewed 2026-06-27 13:56 UTC · model grok-4.3

classification 🧮 math.CV
keywords exponential starlike functionsinverse logarithmic coefficientsHankel determinantHermitian-Toeplitz determinantFekete-Szegő functionalsubordinationcoefficient estimatesanalytic functions
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The pith

The exponential starlike class admits sharp bounds on inverse logarithmic coefficients, associated determinants, and the generalized Fekete-Szegő functional.

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

The paper studies the class of normalized analytic starlike functions where the logarithmic derivative is subordinate to an exponential with parameter alpha between 0 and 1. It derives sharp upper bounds for the first three inverse logarithmic coefficients and for the difference between the second and first. Sharp estimates are given for a second-order inverse logarithmic Hankel determinant and for a third-order Hermitian-Toeplitz determinant. The work also completely solves the extremal problem for the expression involving a3 minus lambda times a2 squared, minus mu times the absolute value of a2. These results supply exact information on coefficient behavior that controls growth and distortion properties in this subclass.

Core claim

For functions f in the class S_ex^* defined by zf'(z)/f(z) subordinate to e^{αz} for 0<α≤1, sharp upper bounds are determined for the inverse logarithmic coefficients Γ1, Γ2, and Γ3; sharp upper and lower bounds are established for |Γ2|−|Γ1|; sharp estimates are derived for the second-order inverse logarithmic Hankel determinant H_{2,1}(F_{f^{-1}}/2); sharp upper and lower bounds are obtained for the third-order Hermitian-Toeplitz determinant T_{3,1}(f); and the extremal problem for the generalized Fekete-Szegő functional |a3−λa2²|−μ|a2| is completely solved, with all estimates shown to be sharp and attained by explicit extremal functions.

What carries the argument

The subordination relation zf'(z)/f(z) ≺ e^{αz} for 0<α≤1, which defines the class and enables direct use of coefficient inequalities from subordination theory.

If this is right

  • The bounds are attained by explicit extremal functions, allowing direct verification for each problem considered.
  • The results resolve the extremal values of the generalized Fekete-Szegő expression for arbitrary parameters λ and μ.
  • The estimates on the determinants constrain the possible coefficient growth within the entire class.
  • Sharpness holds uniformly for the full interval 0<α≤1, including the boundary case α=1.

Where Pith is reading between the lines

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

  • The same subordination approach could be tested on related classes defined by other subordinations to obtain comparative coefficient sizes.
  • The explicit bounds might support numerical checks of whether the class satisfies additional geometric inequalities not addressed in the paper.
  • If the bounds are correct, they limit the possible locations of singularities or the range of the inverse function.

Load-bearing premise

That membership in the class defined by subordination to e^{αz} permits direct application of standard coefficient inequalities from subordination theory without further restrictions on the range of α.

What would settle it

A function satisfying zf'(z)/f(z) ≺ e^{αz} for some α in (0,1] but for which one of the stated sharp bounds on Γ1, Γ2, Γ3 or the Fekete-Szegő expression is violated.

Figures

Figures reproduced from arXiv: 2606.10186 by Nabadwip Sarkar, Pradip Das.

Figure 1
Figure 1. Figure 1: Conformal evolution of the open unit disk D onto the target family ϕ(D) under nested parameter variations of α ∈ {0.5, 0.75, 1.0}. 1.1. Inverse Logarithmic Coefficients. For a function f ∈ S, the inverse logarithmic coefficients Γn, introduced by Ponnusamy et al. [40], are defined by the inverse function f −1 through the expansion Ff−1 (w) := log f −1 (w) w = 2X∞ n=1 Γnw n , |w| < 1 4 . (1.7) [PITH_FULL_I… view at source ↗
read the original abstract

In this paper, we investigate several classical coefficient problems for the geometric subclass $\mathcal{S}_{ex}^{\ast}$ of normalized analytic starlike functions defined by the exponential subordination condition \[ \frac{zf'(z)}{f(z)} \prec e^{\alpha z}, \qquad 0 < \alpha \le 1. \] We determine sharp upper bounds for the initial inverse logarithmic coefficients $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$, and establish sharp upper and lower bounds for the consecutive difference $|\Gamma_2| - |\Gamma_1|$. Furthermore, we derive sharp estimates for the second-order inverse logarithmic Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$ and obtain sharp upper and lower bounds for the third-order Hermitian--Toeplitz determinant $T_{3,1}(f)$. Finally, we provide a complete solution to the extremal problem for the generalized Fekete--Szeg\H{o} functional \[ |a_3 - \lambda a_2^2| - \mu |a_2|. \] In each problem considered, the obtained estimates are shown to be sharp, and the corresponding extremal functions are explicitly characterized.

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.

Referee Report

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Summary. The manuscript investigates several coefficient problems for the geometric subclass S_ex^* of normalized analytic starlike functions defined by the subordination condition zf'(z)/f(z) ≺ e^{αz} for 0<α≤1. It determines sharp upper bounds for the initial inverse logarithmic coefficients Γ1, Γ2, and Γ3; establishes sharp upper and lower bounds for the consecutive difference |Γ2|−|Γ1|; derives sharp estimates for the second-order inverse logarithmic Hankel determinant H_{2,1}(F_{f^{-1}}/2); obtains sharp upper and lower bounds for the third-order Hermitian–Toeplitz determinant T_{3,1}(f); and provides a complete solution to the extremal problem for the generalized Fekete–Szegő functional |a3−λa2²|−μ|a2|. In each case the estimates are asserted to be sharp with explicitly characterized extremal functions.

Significance. If the derivations hold, the paper supplies sharp coefficient bounds and determinant estimates in a specific exponential-subordination subclass of starlike functions, extending classical subordination techniques. The explicit construction of extremal functions (via ω(z)=z and the associated ODE) and the parameter-dependent resolution of the generalized Fekete–Szegő problem constitute concrete strengths that can serve as references for further work on coefficient problems in geometric function theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for recommending acceptance. The referee's summary accurately captures the contributions of the paper.

Circularity Check

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No significant circularity detected

full rationale

The paper defines the class via the subordination zf'(z)/f(z) ≺ e^{αz} (0<α≤1) and applies standard coefficient lemmas from subordination theory to extract bounds on inverse logarithmic coefficients Γ_k, Hankel and Hermitian-Toeplitz determinants, and the generalized Fekete-Szegő functional. Extremal functions are constructed explicitly by setting ω(z)=z in p(z)=e^{α ω(z)} and solving the resulting ODE; all estimates are obtained via direct recursion and case analysis on parameters without any reduction of a 'prediction' to a fitted input, self-definition, or load-bearing self-citation chain. The central claims remain independent of the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard axioms of analytic function theory in the unit disk together with the definition of subordination and the known coefficient comparison properties that subordination implies.

axioms (2)
  • standard math Normalized analytic functions in the unit disk satisfy the usual Taylor expansion and growth properties.
    Invoked implicitly by the normalization and the coefficient problems considered.
  • domain assumption Subordination f ≺ g implies |a_n| ≤ |b_n| for the corresponding coefficients.
    This is the key comparison tool used to obtain the stated bounds.

pith-pipeline@v0.9.1-grok · 5750 in / 1467 out tokens · 36514 ms · 2026-06-27T13:56:40.603131+00:00 · methodology

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