Abstract: In this paper, we first establish several sharp inequalities of homogeneous expansion for biholomorphic quasi-convex mappings of type B and order α on the unit ball E in a complex Banach space X by applying the method and technique of complex analysis. Then, as the application of these sharp inequalities, we derive the sharp estimate of third homogeneous expansions for the above mappings defined on the unit polydisk Uⁿ in Cⁿ.

Key words: quasi-convex mappings of type B and order α, inequalities of homogeneous expansions, the sharp estimate of third homogeneous expansions