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

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.

• The paper establishes sharp bounds with explicit extremal functions, proving |H2(2)| ≤ 1/4 and |H3(1)| ≤ 1/9 for the chosen Ma–Minda starlike subclass.
• It employs advanced analytic techniques and Bernstein polynomial methods to derive precise coefficient estimates under quadratic polynomial constraints.
• The findings offer critical insights into coefficient estimation in geometric function theory and suggest further applications in conformal mapping and analytic signal distortion.

Sharp Hankel Determinants in the Starlike Ma–Minda Subclass Associated with Quadratic Polynomials

Introduction and Problem Formulation

This paper addresses the extremal properties of the second and third Hankel determinants in the Ma–Minda starlike subclass of normalized analytic functions, specifically for the subclass associated with the quadratic polynomial $\varphi(z) = 1 + z + \frac{m}{n} z^2$, under the condition $2m \le n$ with $m, n\in \mathbb{N}$. Functions $f$ considered belong to $\mathcal{A}$ (analytic in the unit disk $\mathbb{D}$, $f(0)=0$, $f'(0)=1$), and the subclass is defined by the subordination condition

$\frac{z f'(z)}{f(z)} \prec \varphi(z),$

meaning there exists a Schwarz function $w$ such that $2m \le n$0. The study aligns with the broader geometric function theory and the Ma–Minda unification of starlike and convex function subclasses via analytic subordination.

Admissibility and Starlikeness of $2m \le n$1

The admissibility of $2m \le n$2 as a Ma–Minda function is rigorously analyzed. By expressing $2m \le n$3 with $2m \le n$4, the paper proves:

• Univalence: $2m \le n$5 is univalent in $2m \le n$6 if and only if $2m \le n$7, i.e., $2m \le n$8.
• Starlikeness with respect to $2m \le n$9: Demonstrated via analysis of $m, n\in \mathbb{N}$0 where $m, n\in \mathbb{N}$1, requiring $m, n\in \mathbb{N}$2 for starlikeness.
• Positivity of Real Part: The real part $m, n\in \mathbb{N}$3 for $m, n\in \mathbb{N}$4 is proved for all $m, n\in \mathbb{N}$5, which subsumes the $m, n\in \mathbb{N}$6 range.

The analysis is detailed and includes both boundary and internal disk behavior, verifying all the Ma–Minda subclass requirements under the stated parameter constraints.

Hankel Determinants—Background and Importance

Hankel determinants of coefficients $m, n\in \mathbb{N}$7, and in particular $m, n\in \mathbb{N}$8 and $m, n\in \mathbb{N}$9, provide information about the growth, distortion, and geometric structure of analytic and univalent functions. Sharp bounds for these determinants are significant in the theory of univalent functions, linking to coefficient problems and operator-theoretic questions.

Main Theorems: Sharp Bounds for $f$0 and $f$1

1. Second Hankel Determinant $f$2

• Theorem: For $f$3, $f$4, and the bound is sharp.
• Proof Strategy:
  • Coefficients $f$5 are expressed in terms of the coefficients of a related Schwarz function using the parametric representations from the Carathéodory class.
  • Sharpness is established via explicit construction of an extremal function: for $f$6,
  • $f$7, yielding $f$8.
  • The proof employs technical estimates, including manipulation of Bernstein polynomial representations and extremality arguments.

2. Third Hankel Determinant $f$9

• Theorem: For $\mathcal{A}$0, $\mathcal{A}$1, and the bound is sharp.
• Proof Strategy:
  • The computation follows parallel lines to the $\mathcal{A}$2 case, but is more technical, involving higher-order terms and more involved use of Bernstein polynomial bounds, tensor-product expansions, and careful partitioning of the parameter space to control the maximum.
  • The sharpness for $\mathcal{A}$3 is shown explicitly: for $\mathcal{A}$4,
  • $\mathcal{A}$5, which gives $\mathcal{A}$6.
  • The proof utilizes the properties of the Carathéodory class, Schwarz functions, and established optimization procedures (including Bernstein coefficient analysis and precise bounding arguments within sub-boxes of parameter space).

Both determinantal results not only provide extremal bounds but also rigorously establish the sharpness and identify the extremal functions.

Theoretical and Practical Implications

This paper resolves the precise upper bounds for second and third Hankel determinants in a well-structured Ma–Minda starlike subclass, filling a recognized gap where only partial or less sharp results were previously available.

From a theoretical perspective, the results:

• Give new extremal data for subclasses of univalent functions tied to symmetric quadratic mappings.
• Extend the understanding of coefficient correlation structures in analytic function classes.
• Demonstrate the effectiveness of Bernstein polynomial techniques in bounding multivariate coefficient functionals.

Practical implications include applications in the geometric function theory, specifically in the estimate of various coefficient-related functionals for subclasses important in conformal mapping and complex dynamical systems, where knowledge about the bounds of Hankel determinants has relevance.

Future Directions

Potential avenues for development include:

• Extending the analysis to more general Ma–Minda subclasses with higher-degree polynomials or non-polynomial $\mathcal{A}$7, to investigate possible new extremal phenomena.
• Application of these sharp coefficient bounds in problems of geometric mapping, approximation theory (e.g., Bieberbach-type conjectures), or control of distortion for analytic signals.
• Numerical refinement and automated symbolic computation for handling ever more complex subordination classes in geometric function theory.

Conclusion

The paper achieves the precise determination of the sharp bounds for $\mathcal{A}$8 and $\mathcal{A}$9 in the starlike Ma–Minda subclass associated to quadratic polynomials, identifying extremal functions and consolidating the role of analytic, geometric, and combinatorial techniques in coefficient estimate theory. These results refine the landscape of determinantal inequalities for analytic functions and provide benchmarks for further studies in this active area of geometric complex analysis.