Zalcman Conjecture for Starlike Mappings in Higher Dimensions
Surya Giri
TL;DR
The paper addresses the Zalcman conjecture for higher dimensions by focusing on the case $n=3$ on two canonical domains: the unit ball $\mathbb{B}$ in a complex Banach space and the unit polydisk $\mathbb{U}^n$ in $\mathbb{C}^n$. It uses the normalized locally biholomorphic framework, the starlike class $\mathcal{S}^*(\mathbb{B})$, and the coefficient bounds in the Carathéodory class $\mathcal{P}$ (via the Cho–Rav lemma) to derive a sharp bound for the Zalcman functional when $F(z)=z f(z)\in \mathcal{S}^*(\mathbb{B})$, namely $\left| \left( \frac{ l_z (D^3 F(0) (z^3))}{3! \|z\|^3} \right)^2 - \left( \frac{ l_z (D^5 F(0) (z^5))}{5! \|z\|^5} \right) \right| \le 4$, and proves sharpness via an extremal $\tilde{F}$. It also extends the argument to the unit polydisk, obtaining corresponding sharp inequalities and an explicit extremal attaining equality, thereby providing a partial confirmation of the Zalcman conjecture in dimension three, contributing to higher-dimensional coefficient problems.
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 $\mathbb{C}^n$. These results confirm the validity of the Zalcman conjecture in higher dimensions for $n=3$.