Zalcman Conjecture for Starlike Mappings in Higher Dimensions

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 $\mathbb{C}^n$. These results confirm the validity of the Zalcman conjecture in higher dimensions for $n=3$.

💡 Research Summary

The paper addresses the Zalcman conjecture, a celebrated coefficient problem in geometric function theory, in the setting of several complex variables. The classical conjecture states that for a normalized univalent function (f(z)=z+\sum_{n\ge2}a_nz^n) in the unit disc, the inequality (|a_n^2-a_{2n-1}|\le (n-1)^2) holds for every integer (n\ge2). While the conjecture has been proved for (n=2,3,4,5,6) in one complex variable, its validity in higher dimensions remains largely unexplored.

The author focuses on the case (n=3) and on two natural families of multivariate holomorphic maps: (i) normalized biholomorphic mappings from the unit ball (B) of a complex Banach space (X) into (X) that are starlike, and (ii) normalized biholomorphic mappings from the unit polydisk (U_n\subset\mathbb{C}^n) that are starlike. A mapping (f) is called starlike if for every non‑zero point (z) there exists a norm‑preserving linear functional (l_z) (guaranteed by Hahn–Banach) such that (\operatorname{Re},l_z(Df(z)^{-1}f(z))>0). In the Euclidean case this reduces to the familiar condition (\operatorname{Re}\sum_{k=1}^n f_k(z)z_k>0).

The main results are two sharp estimates for the Zalcman functional associated with (n=3). For a holomorphic scalar function (f) on the ball with (f(0)=1) and (F(z)=zf(z)) belonging to the starlike class (S^*(B)), the following inequality holds for every non‑zero (z\in B): \