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

In this paper, we introduce the concept of biholomorphic convex mapping of order αα on the unit ball BB in a complex Hilbert space XX. Then we provide some sufficient conditions that a locally biholomorphic mapping ff is a biholomorphic convex mapping of order αα; we give an Alexander type theorem between the subclass of convex mappings and the subclass of starlike mappings on BB in a Hilbert space. We also study the order of starlikeness of biholomorphic convex mappings of order αα on BB in a Hilbert space. Finally, we construct some examples of biholomorphic convex mappings of order αα on BB in a Hilbert space by a linear operator.