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