I Criteria for starlikeness for holomorphic mappings.- II Criteria for convexity for holomorphic mappings.- III The growth theorem for holomorphic starlike mappings.- IV The growth theorem for holomorphic convex mappings.- V The distortion theorem for the linearinvariant family.- VI The distortion theorem for holomorphic convex and starlike mappings.- VII The geometrical properties for holomorphic convex mappings on the unit ball.- References.- List of symbols.