The second and third Hankel determinants for certain convex subclass of functions

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 the present paper, we consider $\mathcal{C}(\varphi) := \left{ f \in \mathcal{A} : 1+zf''(z)/f'(z) \prec \varphi(z):=(1+z/2)² \right}$, as subclass of convex functions and compute the sharp second and third Hankel determinants for functions in $\mathcal{C}(\varphi)$.

• The paper introduces sharp bounds for the second and third Hankel determinants, with maximum values of 1/36 and 1/144 respectively, in the Ma–Minda convex subclass.
• It employs advanced coefficient parameterization and optimization techniques through the use of Schwarz function representations.
• The results offer key insights for extremal problems in geometric function theory and set benchmarks for further exploration of analytic functions.

Sharp Estimates of the Second and Third Hankel Determinants for a Ma–Minda Convex Subclass

Introduction and Motivation

The paper focuses on the problem of determining sharp bounds for the second and third Hankel determinants for the Ma–Minda convex subclass $\mathcal{C}(\varphi)$, where $\varphi(z) = (1+z/2)^2$. The class consists of analytic functions $f$ normalized by $f(0) = 0$ and $f'(0) = 1$ in the unit disk $\mathbb{D}$ and satisfying the subordination condition $1 + z f''(z)/f'(z) \prec \varphi(z)$. The framework generalizes classical families in geometric function theory and enables a unified approach to coefficient problems.

Hankel determinants of univalent and convex functions provide fine-grained information about the interrelation of their Taylor coefficients and have implications for growth, extremal problems, and the geometry of associated image domains. Prior research has addressed Hankel bounds for a spectrum of subclasses, but sharp results for the targeted Ma–Minda convex class generated by the specified $\varphi$ were not previously available.

Properties of the Ma–Minda Convex Class

The paper establishes that $\varphi(z) = (1+z/2)^2$ fulfills all required Ma–Minda conditions: holomorphicity, normalization $\varphi(0)=1$, real part positivity, symmetry, and univalence. The domain of $\varphi(z) = (1+z/2)^2$0 is proved to be starlike and to map $\varphi(z) = (1+z/2)^2$1 univalently onto a domain maintaining these properties.

Any $\varphi(z) = (1+z/2)^2$2 admits an explicit integral representation involving a Schwarz function $\varphi(z) = (1+z/2)^2$3: $\varphi(z) = (1+z/2)^2$4 with further reductions when specializations such as $\varphi(z) = (1+z/2)^2$5 or $\varphi(z) = (1+z/2)^2$6 are made.

Hankel Determinants: Definitions and Coefficient Framework

The $\varphi(z) = (1+z/2)^2$7th Hankel determinant for a function $\varphi(z) = (1+z/2)^2$8 is

$\varphi(z) = (1+z/2)^2$9

with well-known specializations:

• The second Hankel determinant: $f$0.
• The third Hankel determinant: $f$1 as a $f$2 determinant depending on $f$3.

Coefficient extraction for $f$4 is performed by expressing $f$5 via the associated Carathéodory and Schwarz function expansions, allowing parametric representations of higher coefficients in terms of $f$6, $f$7, $f$8, and $f$9 (with $f(0) = 0$0 and $f(0) = 0$1).

Main Results: Sharp Determinant Estimates

Second Hankel Determinant

It is shown that for every $f(0) = 0$2,

$f(0) = 0$3

with the estimate proven to be sharp.

The proof employs both algebraic manipulation using the coefficient framework and reduction to a maximization problem over allowed parameter ranges, using convexity and dominance arguments (including application of results from Choi et al. and properties of Carathéodory and Schwarz classes). The extremal function achieving the bound is generated by $f(0) = 0$4.

Third Hankel Determinant

For the same class, the paper establishes that

$f(0) = 0$5

and sharpness is demonstrated.

The derivation requires a detailed parameterization of all coefficients up to $f(0) = 0$6 as functions of the Schwarz function parameters. The bounding of the determinant involves expressing it as a polynomial in these parameters and maximizing using the Bernstein basis representation, thus securing bounds across the entire parameter domain. The sharp value is attained by the function associated with $f(0) = 0$7.

Numerical Extremality and Methodology

Both determinant bounds are shown to be achieved by explicit extremal choices of the Schwarz function $f(0) = 0$8. The proofs invoke advanced combinatorial and algebraic maximization on polydisks, Bernstein basis techniques, and properties of analytic and univalent function theory.

Specifically, the results are:

• For $f(0) = 0$9, the maximal value $f'(0) = 1$0 is realized for the function $f'(0) = 1$1 corresponding to $f'(0) = 1$2.
• For $f'(0) = 1$3, the bound $f'(0) = 1$4 is sharp and realized for $f'(0) = 1$5.

Theoretical and Practical Implications

These sharp coefficient bounds for the Ma–Minda convex class generated by a specific $f'(0) = 1$6 enrich the understanding of the structure of analytic and convex functions. They provide benchmark values for extremal problems in geometric function theory, contribute to the theory of coefficient functionals, and suggest directions for further study of higher-order Hankel determinants for more general Ma–Minda or other special subclasses. The algebraic techniques developed and employed are of interest for broader analytic function and optimization theory.

These results extend the catalogue of sharp quantitative information on the Taylor coefficients of univalent and convex functions, which are central in applications in geometric function theory, extremal problems, and analytic dynamics.

Possible future directions include analogous sharp estimates for other subclasses defined by different generating functions $f'(0) = 1$7, investigation of Hankel determinants of even higher order (e.g., $f'(0) = 1$8), and exploration of connections to other geometric functionals.

Conclusion

The paper successfully establishes the sharp upper bounds for the second and third Hankel determinants for the Ma–Minda convex subclass associated with $f'(0) = 1$9. The use of advanced parametric representation, algebraic manipulation, and geometric convexity arguments yields precise quantification of the determinant functionals. The demonstrated methodology and results have significant implications for the analysis of analytic, univalent, and convex function spaces, particularly within the generalized Ma–Minda framework.