  In this paper, we first introduce the concept of biholomorphic convex mapping
of order alpha on the unit ball B in a complex Hilbert space X. Next we provide
some sufficient conditions that a locally biholomorphic mapping f is a
biholomorphic convex mapping of order alpha and give an Alexander's theorem
between the subclass of convex mappings and the subclass of starlike mappings
on B in Hilbert space. We also obtain the order of starlikeness of
biholomorphic convex mappings of order alpha on B in Hilbert spaces. Finally,
we construct some concrete examples of biholomorphic convex mappings of order
alpha on B in Hilbert spaces by means of a linear operator.
