Some results of quasi-convex mappings which have a \(\varvec{\Phi }\)-parametric representation in higher dimensions

Let \(\mathbf {E_{\mathbb {X}}}\) be a unit ball on complex Banach space \(\mathbb {X}\) and \(\Phi \) be a convex function such that \(\Phi (0)=1\) and \(\Re \Phi (\xi )>0\) on \(\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}\). In this paper, we continue the work related to the class \(Q_\textbf{B}^{\Phi }(\mathbf {E_{\mathbb {X}}})\) of quasi-convex mappings of type \(\textbf{B}\) which have a \(\Phi \)-parametric representation on \(\mathbf {E_{\mathbb {X}}}\), where the mappings \(f\in Q_\textbf{B}^{\Phi }(\mathbf {E_{\mathbb {X}}})\) are k-fold symmetric, \(k\in \mathbb {N}.\) We give the improved Fekete-Szegö inequalities for the class \(Q_\textbf{B}^{\Phi }(\mathbf {E_{\mathbb {X}}})\) and establish the sharp bounds of all terms of homogeneous polynomial expansions for some subclasses of \(Q_\textbf{B}^{\Phi }(\mathbf {E_{\mathbb {X}}})\). Our main results are closely related to the Bieberbach conjecture in higher dimensions.