Bieberbach–Sheng coefficient conjecture for normalized biholomorphic starlike mappings

Prove that every normalized biholomorphic starlike mapping F from the polydisk \(\mathbb{U}^n\) into \(\mathbb{C}^n\) satisfies \(\|D^mF(0)(z^m)\|/m!\leq m\|z\|^m\) for every \(z\in\mathbb{U}^n\) and every integer \(m\geq 2\).

Background

The paper recalls a conjecture posed by Sheng Gong in 1998 for normalized biholomorphic starlike mappings on the unit polydisk. The conjectured estimate is a dimension-corrected analogue of the one-variable Bieberbach bound and concerns all homogeneous derivative orders m≥2m\geq 2.

The authors note that the literal higher-dimensional Bieberbach analogue was disproved by Roper and Suffridge, while the stated conjecture has been confirmed only for the case m=2m=2. Thus, the general assertion for arbitrary m≥2m\geq 2 remains unresolved in the context presented.

References

In 1998, Sheng Gong posed the following conjecture. If $f:\mathbb{U}n\to \mathbb{C}n$ is a normalized biholomorphic starlike mapping, then

\frac{|Dmf(0)(zm)|}{m!}\leq m|z|m,\quad z\in\mathbb{U}n,\quad m=2,3,\ldots

— Sharp Robertson-type coefficient estimates for radial starlike and quasi-convex of type B mappings of order α in complex Banach spaces  (2610.13869 - Majumder et al., 9 Oct 2026) in Introduction, Conjecture A (Bieberbach–Sheng Conjecture)