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

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