Some results of quasi-convex mappings which have a Φ-parametric representation in higher dimensions

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