Toeplitz determinant and generalized Zalcman conjecture for subclasses of starlike mappings in higher dimensions

Abstract: In this manuscript, we establish sharp bounds of the third-order Toeplitz determinant and a particular case of the generalized Zalcman functional for a class of holomorphic mappings defined on the unit ball in a complex Banach space. The obtained estimates yield corresponding bounds for several subclasses of starlike mappings as special cases and also provide higher-dimensional extensions of certain known results from the classical one-dimensional theory.

• The paper establishes sharp estimates for the third-order Toeplitz determinant and the $(2,3)$ generalized Zalcman functional for holomorphic mappings on complex Banach-space unit balls, recovering the one-dimensional constants $84$ and $2$ for starlike mappings.
• The authors reduce the infinite-dimensional coefficient problem to Schwarz-function estimates on the unit disk using supporting functionals, subordination to a function $\Phi$ with positive real part, and Prokhorov–Szynal inequalities.
• The results are dimension-free and attained by explicit extremal mappings, while sharp bounds remain open for complementary parameter ranges and for generalized Zalcman functionals with index pairs beyond $(2,3)$.

This paper establishes sharp estimates for two coefficient functionals—the third-order Toeplitz determinant $T_{3,2}$ and the generalized Zalcman functional $A_2A_3 - A_4$—for a broad class of holomorphic mappings on the unit ball of a complex Banach space. The results are formulated in terms of the coefficients of a univalent function $\Phi$ with $\operatorname{Re}\Phi > 0$, so that the standard subclasses of starlike mappings (starlike, starlike of order $\alpha$, strongly starlike of order $\beta$) follow as special cases. In doing so, the paper extends to several complex variables the classical one-dimensional bounds $\lvert T_{3,2}\rvert \le 84$ for starlike functions due to Ali, Thomas and Vasudevarao, and $\lvert a_2a_3 - a_4\rvert \le 2$ due to Ma.

Setting and class of mappings

Let $X$ be a complex Banach space with unit ball $\mathbb{B}$, and let $A_2A_3 - A_4$0 denote the set of norm-one functionals supporting $A_2A_3 - A_4$1 (nonempty by the Hahn–Banach theorem). For a holomorphic $A_2A_3 - A_4$2 with $A_2A_3 - A_4$3, $A_2A_3 - A_4$4, positive derivative at the origin, and $A_2A_3 - A_4$5 symmetric about the real axis, the class $A_2A_3 - A_4$6 of Graham–Hamada–Kohr consists of mappings $A_2A_3 - A_4$7 with $A_2A_3 - A_4$8, $A_2A_3 - A_4$9, and $\Phi$0. Writing

$\Phi$1

the symmetry assumption forces all $\Phi$2 to be real. The choices $\Phi$3, $\Phi$4, and $\Phi$5 recover, respectively, the starlike mappings $\Phi$6, the starlike mappings of order $\Phi$7 in the sense of Hamada–Kohr–Liczberski, and the strongly starlike mappings of order $\Phi$8 of Kohr–Liczberski.

The functional estimates are stated for mappings of the form $\Phi$9, where $\operatorname{Re}\Phi > 0$0 is holomorphic with $\operatorname{Re}\Phi > 0$1 and $\operatorname{Re}\Phi > 0$2, with the coefficients $\operatorname{Re}\Phi > 0$3 read off the homogeneous expansion of $\operatorname{Re}\Phi > 0$4 via supporting functionals:

$\operatorname{Re}\Phi > 0$5

The Toeplitz determinant $\operatorname{Re}\Phi > 0$6

The main result of the paper is the sharp bound

$\operatorname{Re}\Phi > 0$7

subject to the parameter restrictions $\operatorname{Re}\Phi > 0$8 and $\operatorname{Re}\Phi > 0$9, where $\alpha$0 are explicit functions of $\alpha$1 and the regions $\alpha$2 partition $\alpha$3 in the Prokhorov–Szynal analysis of the Fekete–Szegő-type functional $\alpha$4 over Schwarz functions.

The proof proceeds by reducing the infinite-dimensional problem to the classical disk. For fixed $\alpha$5 and $\alpha$6, the scalar function $\alpha$7 is subordinate to $\alpha$8. The key algebraic identity, obtained from the inverse-derivative formula for $\alpha$9, is

$\beta$0

which links the coefficients of $\beta$1 to the Fréchet derivatives of $\beta$2 at the origin. This yields expressions for $\beta$3, $\beta$4, and $\beta$5 in terms of $\beta$6, $\beta$7, and $\beta$8, so that the Toeplitz determinant factors as $\beta$9. The two factors are then bounded separately using Prokhorov–Szynal coefficient estimates for subordinate functions, in the regions of the $\lvert T_{3,2}\rvert \le 84$0- and $\lvert T_{3,2}\rvert \le 84$1-planes where the bound $\lvert T_{3,2}\rvert \le 84$2 applies.

Sharpness is verified by the explicit extremal mapping

$\lvert T_{3,2}\rvert \le 84$3

for which the homogeneous coefficients are purely alternating in phase and the bound is attained with equality. This is a notable feature: the estimate is not merely an upper bound obtained by crude triangle-inequality estimates, but is attained by a concrete mapping in the class.

Specializing $\lvert T_{3,2}\rvert \le 84$4 gives the following corollaries, each sharp:

Class	Bound on $\lvert T_{3,2}\rvert \le 84$5
$\lvert T_{3,2}\rvert \le 84$6	$\lvert T_{3,2}\rvert \le 84$7
$\lvert T_{3,2}\rvert \le 84$8, $\lvert T_{3,2}\rvert \le 84$9	$\lvert a_2a_3 - a_4\rvert \le 2$0, with $\lvert a_2a_3 - a_4\rvert \le 2$1
$\lvert a_2a_3 - a_4\rvert \le 2$2, $\lvert a_2a_3 - a_4\rvert \le 2$3	$\lvert a_2a_3 - a_4\rvert \le 2$4, with $\lvert a_2a_3 - a_4\rvert \le 2$5

The bound $\lvert a_2a_3 - a_4\rvert \le 2$6 for starlike mappings coincides with the one-dimensional value in Theorem A of Ali–Thomas–Vasudevarao, so the higher-dimensional extension is dimension-free and sharp. The restrictions $\lvert a_2a_3 - a_4\rvert \le 2$7 and $\lvert a_2a_3 - a_4\rvert \le 2$8 arise precisely from the requirement that the corresponding $\lvert a_2a_3 - a_4\rvert \le 2$9 fall in the regions $X$0–$X$1 where the Prokhorov–Szynal bound $X$2 is valid; the paper does not treat the complementary parameter ranges, where a different branch of the coefficient estimate would be needed.

The generalized Zalcman functional

The second main result addresses the case $X$3, $X$4 of Ma's generalized Zalcman conjecture. For the same class of mappings, the paper proves

$X$5

with $X$6 and $X$7. The identity $X$8 reduces the problem to the same Schwarz-function functional as before, now handled across all three branches of the Prokhorov–Szynal estimate. Sharpness is established separately on each branch, using the extremals $X$9 (built from $\mathbb{B}$0, attaining $\mathbb{B}$1), $\mathbb{B}$2 (built from $\mathbb{B}$3), and $\mathbb{B}$4 (built from $\mathbb{B}$5 with $\mathbb{B}$6, the extremal for the intermediate region identified by Cho, Kwon, Lecko and Sim).

For the concrete subclasses, this yields the sharp bounds:

Class	Bound on $\mathbb{B}$7
$\mathbb{B}$8	$\mathbb{B}$9
$A_2A_3 - A_4$00	piecewise: $A_2A_3 - A_4$01 for $A_2A_3 - A_4$02; $A_2A_3 - A_4$03 for $A_2A_3 - A_4$04, $A_2A_3 - A_4$05
$A_2A_3 - A_4$06	three-branch piecewise bound in $A_2A_3 - A_4$07 with breakpoints $A_2A_3 - A_4$08 (root of $A_2A_3 - A_4$09) and $A_2A_3 - A_4$10

The value $A_2A_3 - A_4$11 for starlike mappings recovers Ma's one-dimensional Theorem B in every dimension, and each piece of the piecewise estimates is attained by an explicit extremal mapping. The three-branch structure of the strongly starlike case reflects the full trichotomy of the Prokhorov–Szynal functional, and the paper identifies which extremal mapping corresponds to each parameter range.

Scope and open questions

Two limitations should be noted. First, the Toeplitz result is proved only under the parameter restrictions $A_2A_3 - A_4$12 and $A_2A_3 - A_4$13; consequently the corollaries for $A_2A_3 - A_4$14 and $A_2A_3 - A_4$15 cover only the stated subintervals of $A_2A_3 - A_4$16 and $A_2A_3 - A_4$17, and sharp bounds on the complementary ranges remain open. Second, the Zalcman-type result, like its one-dimensional antecedent, covers only the pair $A_2A_3 - A_4$18; the generalized Zalcman conjecture for general pairs in higher dimensions is not addressed. The method also requires the symmetry of $A_2A_3 - A_4$19 about the real axis (ensuring real $A_2A_3 - A_4$20), so Ma–Minda classes built from non-symmetric domains fall outside the present framework.

Conclusion

The paper supplies the first sharp estimates of the third-order Toeplitpitz determinant $A_2A_3 - A_4$21 and of the $A_2A_3 - A_4$22-case of the generalized Zalcman functional for holomorphic mappings on the unit ball of a complex Banach space, uniformly over the $A_2A_3 - A_4$23 framework. The bounds are dimension-independent, reduce exactly to the known sharp one-dimensional constants ($A_2A_3 - A_4$24 and $A_2A_3 - A_4$25) for starlike mappings, and are attained by explicit extremals on each branch of the parameter space. The principal open problems left by the work are the sharp $A_2A_3 - A_4$26 bound on the complementary parameter ranges of $A_2A_3 - A_4$27 and $A_2A_3 - A_4$28, and extensions of the generalized Zalcman estimate to arbitrary index pairs in higher dimensions.