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In this paper, we first obtain several sharp inequalities of homogeneousexpansion for both the subclass of all normalized biholomorphic quasi-convexmappings of type B and order alpha and the subclass of all normalizedbiholomorphic almost starlike mappings of order alpha defined on the unit ball$B$ of a complex Banach space X. Then, with these sharp inequalities, we derivethe sharp estimates of the third and fourth homogeneous expansions for theabove mappings defined on the unit polydisk $D^n$ in C(n).

Sharp inequalities of homogeneous expansions for quasi-convex mappings of type B and almost starlike mappings of order alpha