In　this　paper,　a　new　class　of　biholomorphic　mappings　named　＂epsilon　quasi-convex　mapping＂　is　introduced　in　the　unit　ball　of　a　complex　Banach　space.　Meanwhile,　the　definition　of　epsilon-starlike　mapping　is　generalized　from　epsilon　is　an　element　of　[0,　1]　to　epsilon　is　an　element　of　[-1,　1].　It　is　proved　that　the　class　of　epsilon　quasi-convex　mappings　is　a　proper　subset　of　the　class　of　starlike　mappings　and　contains　the　class　of　epsilon　starlike　mappings　properly　for　some　epsilon　is　an　element　of　[-1,　0)　boolean　OR　(0,　1].　We　give　a　geometric　explanation　for　epsilon-starlike　mapping　with　epsilon　is　an　element　of　[-1,　1]　and　prove　that　the　generalized　Roper-Suffridge　extension　operator　preserves　the　biholomorphic　epsilon　starlikeness　on　some　domains　in　Banach　spaces　for　epsilon　is　an　element　of　[-1,　1].　We　also　give　some　concrete　examples　of　epsilon　quasi-convex　mappings　or　e　starlike　mappings　for　epsilon　is　an　element　of　[-1,　1]　in　Banach　spaces　or　C-n.　Furthermore,　some　other　properties　of　epsilon　quasi-convex　mapping　or　epsilon-starlike　mapping　are　obtained.　These　results　generalize　the　related　works　of　some　authors.　(c)　2005　Elsevier　Inc.　All　rights　reserved.