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

In this paper, we introduce the concept of biholomorphic convex mapping of order α 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 α ; 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 α on B in a Hilbert space. Finally, we construct some examples of biholomorphic convex mappings of order α on B in a Hilbert space by a linear operator.