Unified solution of Fekete-Szegö problem for subclasses of starlike mappings in several complex variables

Let $\begin{array}{} \mathcal {S}^*_\psi \end{array}$ be a subclass of starlike functions in the unit disk 𝕌, where ψ is a convex function such that ψ(0) = 1, ψ′(0) > 0, ℜ(ψ(ξ)) > 0 and ψ(𝕌) is symmetric with respect to the real axis. We obtain the sharp solution of Fekete-Szegö problem for the family $\begin{array}{} \mathcal {S}^*_\psi \end{array}$, and then extend the result to the case of corresponding subclass defined on the bounded starlike circular domain Ω in several complex variables, which give an unified answer of Fekete-Szegö problem for the kinds of subclasses of starlike mappings defined on Ω. At last, we propose two conjectures related the same problems on the unit ball in a complex Banach space and on the unit polydisk in ℂn.