Biholomorphic convex mappings of order α on the unit ball in Hilbert spaces

In this paper, we introduce the concept of biholomorphic convex mapping of order a on the unit ball B in a complex Hilbert space X. Then we provide some sufficient conditions that a locally biholomorphic mapping f is a biholomorphic convex mapping of order a; we give an Alexander type theorem between the subclass of convex mappings and the subclass of starlike mappings on B in a Hilbert space. We also study the order of starlikeness of biholomorphic convex mappings of order a on B in a Hilbert space. Finally, we construct some examples of biholomorphic convex mappings of order a on B in a Hilbert space by a linear operator. (C) 2013 Elsevier Inc. All rights reserved.