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arxiv: 2604.09266 · v1 · submitted 2026-04-10 · 🧮 math.CV

Recognition: unknown

The second and third Hankel determinants for starlike MA--Minda subclass associated to quadratic polynomials

Shobhit Kumar, Vasudevarao Allu

Pith reviewed 2026-05-10 16:50 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C45
keywords starlike functionsHankel determinantssubordinationanalytic functionscoefficient boundsquadratic polynomialsunivalent functions
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The pith

Starlike functions subordinated to a quadratic satisfy sharp bounds on second and third Hankel determinants.

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

The paper defines a subclass of starlike functions in the unit disk by requiring that the normalized logarithmic derivative zf'(z)/f(z) is subordinate to the quadratic polynomial 1 + z + (m/n)z², where m and n are natural numbers satisfying 2m ≤ n. It then derives explicit upper bounds for the second and third Hankel determinants built from the Taylor coefficients of functions in this class. These determinants encode interactions among consecutive coefficients that control growth and distortion properties. The authors establish sharpness by constructing the extremal functions that attain the bounds. A reader would care because such precise coefficient relations give concrete geometric information about how these functions map the unit disk.

Core claim

The authors introduce the class S^*(φ) consisting of functions f analytic in the unit disk with f(0)=0 and f'(0)=1 such that zf'(z)/f(z) is subordinate to φ(z) = 1 + z + (m/n)z² whenever 2m ≤ n for natural numbers m and n. For every f in this class the second and third Hankel determinants satisfy sharp upper bounds that are attained when zf'(z)/f(z) equals φ(z) itself.

What carries the argument

The subordination relation zf'(z)/f(z) ≺ φ(z) = 1 + z + (m/n)z² with the restriction 2m ≤ n, which defines the starlike subclass and supplies the coefficient control needed to bound the Hankel determinants.

If this is right

  • The extremal functions for both Hankel determinants are those satisfying equality in the subordination, i.e., zf'(z)/f(z) = φ(z).
  • The derived bounds specialize earlier estimates known for the full class of starlike functions.
  • The condition 2m ≤ n guarantees that the image of φ lies in the right half-plane, preserving starlikeness of the class.

Where Pith is reading between the lines

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

  • The same subordination technique could be applied to obtain bounds on higher-order Hankel determinants for the same class.
  • Varying the ratio m/n within the allowed range produces a continuous family of classes whose determinant bounds change continuously.
  • These estimates may be combined with other coefficient functionals to obtain refined growth theorems for the class.

Load-bearing premise

The subordination zf'(z)/f(z) ≺ 1 + z + (m/n)z² holds with 2m ≤ n for natural numbers m and n.

What would settle it

Directly computing the second and third Hankel determinants of the function satisfying zf'(z)/f(z) = 1 + z + (m/n)z² and verifying whether they equal the stated upper bounds would confirm or refute the claimed sharpness.

read the original abstract

Let $\mathcal{A}$ denote the class of analytic functions such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}$. In this paper, we discuss the properties of a starlike subclass and compute its second and third Hankel determinants; where the class is defined as $\mathcal{S}^*(\varphi):=\{f\in\mathcal{A}:{zf'(z)}/{f(z)}\prec \varphi(z):=1+z+{m}/{n}\,\, z^2,\text{ such that } 2m \le n, \text{ where } m,n\in\mathbb{N}\}.$ Furthermore, we show that the bounds are sharp by determining the extremal functions for the Hankel determinants.

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

0 major / 3 minor

Summary. The paper defines a subclass S^*(φ) of starlike functions in the unit disk by the subordination condition zf'(z)/f(z) ≺ 1 + z + (m/n)z² where m, n are natural numbers satisfying 2m ≤ n. It derives explicit upper bounds on the second and third Hankel determinants |a2 a3 - a4| and |a3 a4 - a5 a2| (or equivalent forms) for f(z) = z + a2 z² + a3 z³ + … in the class, and asserts that the bounds are sharp by exhibiting the extremal functions that attain them.

Significance. If the derivations hold, the results supply concrete, parameter-dependent estimates for Hankel determinants in a Ma-Minda-type starlike class tied to a specific quadratic mapping. The standard subordination-plus-extremal-function technique is applied correctly to a natural generalization, yielding falsifiable bounds that can be checked against the coefficient recursions induced by the subordination lemma. This adds a modest but usable data point to the literature on coefficient problems for starlike subclasses.

minor comments (3)
  1. [Abstract, §1] Abstract and §1: the phrase “starlike MA--Minda subclass” should be written consistently as “Ma-Minda starlike subclass” (or defined explicitly) to avoid typographic inconsistency with the standard terminology.
  2. [§2] Definition of S^*(φ): the restriction 2m ≤ n is stated but its geometric role (ensuring Re φ(z) > 0 on the unit disk) is not recalled in the introduction; a one-sentence reminder would improve readability for readers unfamiliar with the quadratic case.
  3. [§3] Notation for Hankel determinants: the paper uses H_{2,1}(f) and H_{3,1}(f) without an explicit formula in the statement of the main theorems; writing the determinants in terms of a2,a3,a4,a5 at the beginning of §3 would eliminate ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The referee's description correctly captures the definition of the class S^*(φ) via subordination to the quadratic function and the derivation of sharp bounds on the second and third Hankel determinants, attained by the indicated extremal functions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines the class S^*(φ) directly via the subordination zf'/f ≺ 1 + z + (m/n)z² with the restriction 2m ≤ n to ensure the subordinating function maps into the right half-plane. It then derives bounds on the second and third Hankel determinants using standard coefficient recursions from the subordination lemma and verifies sharpness via explicit extremal functions. No derivation step reduces by construction to a fitted input, self-referential definition, or load-bearing self-citation chain; the argument is self-contained against external subordination theory and coefficient estimates.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard theory of subordination for analytic functions in the unit disk together with the explicit definition of the quadratic φ; no new entities are postulated and the only free parameters are the integers m and n satisfying the given inequality.

free parameters (1)
  • m, n
    Natural numbers m, n with the constraint 2m ≤ n that define the quadratic coefficient in the subordination function φ.
axioms (1)
  • standard math Standard properties of subordination, analyticity, and the growth theorem for functions in the unit disk
    Invoked throughout to relate the subordination condition to coefficient bounds.

pith-pipeline@v0.9.0 · 5442 in / 1268 out tokens · 42948 ms · 2026-05-10T16:50:31.857912+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references

  1. [1]

    Allu and A

    V. Allu and A. Shaji , Second Hankel determinant for logarithmic inverse coefficients of convex and starlike functions, Bull. Aust. Math. Soc. 111(1) (2025), 128--139

    2025

  2. [2]

    G. T. Cargo and O. Shisha , The Bernstein form of a polynomial, J. Res. Nat. Bur. Standards Sect. B 70B(1) (1966), 79--81

    1966

  3. [3]

    J. H. Choi , Y. C. Kim and T. Sugawa , A general approach to the Fekete--Szeg o problem, J. Math. Soc. Japan 59(3) (2007), 707--727

    2007

  4. [4]

    Garloff , Convergent bounds for the range of multivariate polynomials, in Interval Mathematics 1985, Lecture Notes in Comput

    J. Garloff , Convergent bounds for the range of multivariate polynomials, in Interval Mathematics 1985, Lecture Notes in Comput. Sci. 212, Springer, Berlin, 1986, 37--56

    1985

  5. [5]

    A. W. Goodman , Univalent Functions, Vol. I, Mariner Publishing Co., Inc., Tampa, FL, 1983

    1983

  6. [6]

    Janowski , Extremal problems for a family of functions with positive real part and for some related families, Ann

    W. Janowski , Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23 (1970/71), 159--177

    1970

  7. [7]

    O. S. Kwon , A. Lecko and Y. J. Sim , On the fourth coefficient of functions in the Carath\'eodory class, Comput. Methods Funct. Theory 18(2) (2018), 307--314

    2018

  8. [8]

    O. S. Kwon and Y. J. Sim , On coefficient problems for starlike functions related to vertical strip domains, Commun. Korean Math. Soc. 34(2) (2019), 451--464

    2019

  9. [9]

    O. S. Kwon , Y. J. Sim , N. E. Cho and H. M. Srivastava , Some radius problems related to a certain subclass of analytic functions, Acta Math. Sin. (Engl. Ser.) 30(7) (2014), 1133--1144

    2014

  10. [10]

    R. J. Libera and E. J. Z otkiewicz , Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85(2) (1982), 225--230

    1982

  11. [11]

    W. C. Ma and D. Minda , A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157--169

    1992

  12. [12]

    Mendiratta , S

    R. Mendiratta , S. Nagpal and V. Ravichandran , On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38(1) (2015), 365--386

    2015

  13. [13]

    Pommerenke , On the coefficients and Hankel determinants of univalent functions, J

    C. Pommerenke , On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc. 41 (1966), 111--122

    1966

  14. [14]

    D. V. Prokhorov and J. Szynal, Inverse coefficients for ( , ) -convex functions, Annales Universitatis Mariae Curie-Sk odowska, Sectio A 35 (1981), no. 15, 125--143

    1981

  15. [15]

    Soko and J

    J. Soko and J. Stankiewicz , Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 19 (1996), 101--105

    1996