On quasi-convex mappings of order α in the unit ball of a complex Banach space

In this paper, a class of biholomorphic mappings named quasi-convex mapping of order α in the unit ball of a complex Banach space is introduced. When the Banach space is confined to ℂ ⁿ , we obtain the relation between this class of mappings and the convex mappings. Furthermore, the growth and covering theorems of this class of mappings are given on the unit ball of a complex Banach space X. Finally, we get the second order terms coefficient estimations of the homogeneous expansion of quasi-convex mapping of order α defined on the polydisc in ℂ ⁿ and on the unit ball in a complex Banach space, respectively. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday