Third-Order Toeplitz Determinant for a Subclass of Starlike Mappings in Higher Dimensions

Abstract: The manuscript establishes sharp bound of the third-order Toeplitz determinant for a subclass of starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in $\mathbb{C}<sup>n.$

• The paper establishes a sharp upper bound of 84 for the third-order Toeplitz determinant using higher-order Fréchet derivatives.
• It generalizes classical one-dimensional starlike function results to biholomorphic mappings on complex Banach spaces and circular domains.
• The work bridges classical geometric function theory with modern multivariate analysis, offering new insights into coefficient problems in several complex variables.

Third-Order Toeplitz Determinant Bounds for Starlike Mappings in Higher Dimensions

Introduction

The paper addresses the sharp upper bound for the modulus of the third-order Toeplitz determinant $|T_{3,2}(f)|$ for a subclass of starlike biholomorphic mappings in the context of several complex variables. The work generalizes established results from the one-dimensional theory—particularly for starlike functions in the unit disk—to analytic mappings defined on the unit ball of a complex Banach space and on bounded starlike circular domains in $\mathbb{C}^n$. In higher dimensions, many classical geometric function theory results do not admit direct generalization, notably due to the breakdown of results like the Bieberbach conjecture. The main contribution is a sharp inequality involving Fréchet derivatives for the third-order Toeplitz determinant, establishing boundaries previously unknown in multidimensional settings.

Theoretical Framework

The study centers on normalized biholomorphic and starlike mappings defined as holomorphic functions with holomorphic inverses (locally or globally) on the unit ball $B$ in a complex Banach space $X$, or on bounded starlike circular domains $\Omega \subset \mathbb{C}^n$. The work employs higher-order Fréchet derivatives to formulate analogs of the coefficient conditions found in the classical univariate theory. Key technical machinery includes the use of the Carathéodory class in higher dimensions, properties of the Minkowski functional for circular domains, and functional-analytic methods such as the Hahn-Banach theorem to ensure the existence and properties of certain linear functionals on $X$.

The Toeplitz determinant of interest, in the notationally consistent setting, is given by

$T_{3,2}(f) = (a_2 - a_4)(a_2 - 2a_3 + a_2 a_4)$

where the $a_j$ are generalized (multi-index or Fréchet) coefficients derived from the holomorphic expansion of mappings on the relevant domain.

Main Results

The primary achievements of the paper are the following two theorems, which establish sharp upper bounds for $|T_{3,2}(f)|$, expressed in terms of higher order Fréchet derivatives.

Theorem 5: Mappings on the Unit Ball in a Complex Banach Space

Let $F(z) = z f(z)$, with $\mathbb{C}^n$0 holomorphic and $\mathbb{C}^n$1 normalized and starlike on $\mathbb{C}^n$2. Define

$\mathbb{C}^n$3

where $\mathbb{C}^n$4 is chosen from the norm-exposing functionals on $\mathbb{C}^n$5. Then:

$\mathbb{C}^n$6

This bound is proved to be sharp; the extremal function is constructed analogously to the one-dimensional extremizer.

Theorem 6: Mappings on Bounded Starlike Circular Domains in $\mathbb{C}^n$7

If $\mathbb{C}^n$8 is defined as above but on a smooth starlike circular domain $\mathbb{C}^n$9 with Minkowski functional $B$0 and the same normalization and starlikeness conditions, the analogous coefficients (involving the directional derivative with respect to $B$1) and the same form for the third-order Toeplitz determinant apply:

$B$2

Again, sharpness is realized by explicit construction.

Comparison With One-Dimensional Theory and Previous Work

The one-variable result, $B$3 for $B$4 in the classical starlike class, is due to Ali et al. The present work extends this via functional analytic generalization and precise analysis of multidimensional expansions. Previous multidimensional results addressed lower-order Toeplitz determinants such as $B$5 and $B$6, but $B$7 remained unbounded until this investigation. The work also inherently relates to coefficient estimates for the Carathéodory class and exploits results on sharp bounds from related classes of holomorphic functions.

Implications and Future Directions

By establishing the sharp bound for $B$8 in higher dimensions, this research rigorously delineates the limit of growth for that class of Toeplitz determinants, contributing to the understanding of coefficient problems in several complex variables. This has ramifications for the broader study of univalent and starlike mapping classes in areas such as multivariate complex analysis, operator theory, and geometric function theory. The techniques developed—especially those involving the coupling of Fréchet derivatives with geometric properties of domains—invite further exploration of coefficient functionals and extremal problems in more complex settings, including other subclasses (convex, close-to-convex, quasi-convex mappings), additional function spaces, or mappings under alternative normalization constraints.

Extensions to more general Banach space-valued analytic function spaces or to domains with less regular Minkowski functionals could present new avenues of research. Open directions also include seeking analogous sharp bounds for other coefficient-based determinants or combinatorial functionals relevant for univalent mapping theory in several variables.

Conclusion

This paper provides a definitive solution for the sharp upper bound of the modulus of the third-order Toeplitz determinant for normalized starlike biholomorphic mappings in higher dimensions, both in abstract Banach spaces and classical domains in $B$9. The results fill a notable gap in the multidimensional geometric function theory literature and offer a blueprint for approaching similar extremal function problems involving analytic mappings and Toeplitz-type determinants. The established sharpness, combined with a flexible analytic approach, lays a foundation for future developments in the theory of holomorphic mappings in complex Banach spaces and several complex variables (2606.17778).