Zalcman Conjecture for Starlike Mappings in Higher Dimensions

Abstract: Counterexamples show that many results in the geometric function theory of one complex variable are not applicable for several complex variables. In this paper, we obtain sharp bounds for the Zalcman functional for $n=3$ associated with the starlike mappings defined on the unit ball in a complex Banach space and on the unit polydisk in $\mathbb{C}<sup>n$. These results confirm the validity of the Zalcman conjecture in higher dimensions for $n=3$.

• The paper extends the Zalcman conjecture’s n=3 functional to starlike mappings in the unit ball of Banach spaces and unit polydisk.
• The derived bounds encompass a polynomial in the Taylor coefficients, maintaining the sharpness constant of 4
• The resulted constructions indicate the robustness of the Zalcman establishment, holding consistently against general context functioning

Background and motivation

The Zalcman conjecture, proposed in 1960 for the class $\mathcal{S}$ of normalized univalent functions on the unit disk $f(z)=z+\sum_{n\ge 2}a_n z^n$, asserts that

$|a_n^2 - a_{2n-1}| \le (n-1)^2,\qquad n \ge 2.$

Brown and Tsao observed that the conjecture implies the Bieberbach conjecture $|a_n|\le n$, and they settled it for starlike and typically real functions; Ma proved it for close-to-convex functions. Krushkal established the cases $n=3,4,5,6$ in full generality, while the case $n\ge 7$ remains open. For starlike functions, the case $n=3$ reduces to the sharp bound $|a_3^2 - a_5|\le 4$.

The situation in several complex variables is structurally different: Cartan showed that the Bieberbach estimate fails for biholomorphic mappings in higher dimensions without additional hypotheses, and Poincaré's work on the failure of the Riemann mapping theorem underscores that one-variable techniques do not transfer automatically. The present paper extends the $n=3$ Zalcman functional to starlike mappings defined on the unit ball $\mathbb{B}$ of a complex Banach space $f(z)=z+\sum_{n\ge 2}a_n z^n$0 and on the unit polydisk $f(z)=z+\sum_{n\ge 2}a_n z^n$1, obtaining sharp bounds with constant $f(z)=z+\sum_{n\ge 2}a_n z^n$2—the same numerical value as in the classical one-variable result.

Setting and definitions

The author works within the standard framework of holomorphic mappings on bounded symmetric domains. A normalized locally biholomorphic mapping $f(z)=z+\sum_{n\ge 2}a_n z^n$3 ($f(z)=z+\sum_{n\ge 2}a_n z^n$4, $f(z)=z+\sum_{n\ge 2}a_n z^n$5) is starlike if

$f(z)=z+\sum_{n\ge 2}a_n z^n$6

where $f(z)=z+\sum_{n\ge 2}a_n z^n$7 denotes the set of norming functionals guaranteed by the Hahn–Banach theorem. On the polydisk this condition specializes to $f(z)=z+\sum_{n\ge 2}a_n z^n$8 componentwise, where $f(z)=z+\sum_{n\ge 2}a_n z^n$9 and $|a_n^2 - a_{2n-1}| \le (n-1)^2,\qquad n \ge 2.$0 is an index with $|a_n^2 - a_{2n-1}| \le (n-1)^2,\qquad n \ge 2.$1. The classes are denoted $|a_n^2 - a_{2n-1}| \le (n-1)^2,\qquad n \ge 2.$2 and $|a_n^2 - a_{2n-1}| \le (n-1)^2,\qquad n \ge 2.$3.

The key reduction device is a scalar Carathéodory function. For fixed $|a_n^2 - a_{2n-1}| \le (n-1)^2,\qquad n \ge 2.$4, define

$|a_n^2 - a_{2n-1}| \le (n-1)^2,\qquad n \ge 2.$5

which satisfies $|a_n^2 - a_{2n-1}| \le (n-1)^2,\qquad n \ge 2.$6 and $|a_n^2 - a_{2n-1}| \le (n-1)^2,\qquad n \ge 2.$7, i.e., $|a_n^2 - a_{2n-1}| \le (n-1)^2,\qquad n \ge 2.$8. Using an inverse-derivative identity of Pfaltzgraff–Suffridge type for mappings of the form $|a_n^2 - a_{2n-1}| \le (n-1)^2,\qquad n \ge 2.$9, the functional coefficients of $|a_n|\le n$0 are expressed in terms of the Fréchet derivatives $|a_n|\le n$1, allowing the Zalcman combination to be rewritten as a polynomial in the Taylor coefficients of $|a_n|\le n$2.

Main results

Banach-space setting. If $|a_n|\le n$3 with $|a_n|\le n$4 and $|a_n|\le n$5, then

$|a_n|\le n$6

and the bound is sharp. The proof expands the Zalcman expression as

$|a_n|\le n$7

then applies the coefficient estimates for Carathéodory functions, namely $|a_n|\le n$8 and $|a_n|\le n$9, together with the triangle inequality. Sharpness is attained by the extremal mapping $n=3,4,5,6$0 with $n=3,4,5,6$1: along the ray $n=3,4,5,6$2 the two relevant normalized derivatives equal $n=3,4,5,6$3 and $n=3,4,5,6$4, giving exactly $n=3,4,5,6$5. This is the direct analogue of the Koebe-type extremal function $n=3,4,5,6$6 in one variable.

Polydisk setting. If $n=3,4,5,6$7, then

$n=3,4,5,6$8

again sharply. The argument is parallel: for each coordinate $n=3,4,5,6$9 with $n\ge 7$0 one constructs a Carathéodory function $n\ge 7$1, derives the pointwise bound $n\ge 7$2 on the distinguished boundary via Lemma (Cho–Kumar–Ravichandran), and then lifts to the full polydisk by the maximum modulus theorem applied to the holomorphic functions obtained from each coordinate of the vector-valued Zalcman expression. The stated extremal function is the coordinatewise analogue $n\ge 7$3, evaluated along $n\ge 7$4.

In both cases, specializing to $n\ge 7$5, $n\ge 7$6 recovers Theorem A of Brown and Tsao, so the results are genuine extensions rather than distinct phenomena. The uniform constant $n\ge 7$7 across dimensions is the salient quantitative finding: the Zalcman functional at $n\ge 7$8 does not deteriorate when passing from one to several complex variables for starlike mappings, in contrast to the failure of unrestricted Bieberbach-type growth estimates noted by Cartan.

Methodological remarks

Two features of the proof deserve emphasis. First, the reduction to a single Carathéodory function per direction converts an infinite-dimensional coefficient problem into the classical one-variable coefficient problem for $n\ge 7$9, whose sharp second-order-type inequalities are known. Second, the structure $n=3$0 with scalar $n=3$1 is essential: it yields the explicit inverse formula $n=3$2, from which all derivative relations follow. The method therefore covers radial perturbations of the identity rather than the full class $n=3$3 of arbitrary starlike mappings—a restriction inherent to the hypotheses of both theorems, which assume $n=3$4 with scalar holomorphic $n=3$5.

Limitations and open questions

Several caveats are explicit or implicit in the paper. The results address only the case $n=3$6 of the Zalcman functional; the corresponding higher-order functionals $n=3$7 in several variables are not treated, and even in one variable the conjecture is open for $n=3$8. Both main theorems require the special form $n=3$9 with $|a_3^2 - a_5|\le 4$0 scalar-valued, so the bounds do not immediately apply to general starlike mappings in $|a_3^2 - a_5|\le 4$1; extending them to the full class is a natural open question. Additionally, the sharpness example in the polydisk theorem contains typographical irregularities in the source (the displayed extremal mapping repeats the index $|a_3^2 - a_5|\le 4$2 across coordinates), though the intended construction—the one-variable Koebe function embedded in the first coordinate—is clear from context. Whether the maximum-modulus lifting argument can be adapted to other domains such as the unit ball of $|a_3^2 - a_5|\le 4$3 with the Euclidean structure, where the distinguished-boundary reduction used here is not available in the same form, also remains unaddressed.

Conclusion

The paper establishes sharp Zalcman-type estimates with constant $|a_3^2 - a_5|\le 4$4 for the $|a_3^2 - a_5|\le 4$5 functional associated with starlike mappings of the form $|a_3^2 - a_5|\le 4$6 on the unit ball of a complex Banach space and on the unit polydisk, thereby extending the Brown–Tsao one-variable theorem to higher dimensions and providing partial confirmation of the Zalcman conjecture beyond the classical setting. The proofs rest on a clean reduction to Carathéodory coefficient bounds, and the sharpness constructions show the constant cannot be improved. The principal unresolved issue raised by the work is whether analogous estimates hold for general starlike mappings without the radial-product hypothesis, and for higher indices of the Zalcman functional.