A boundedly holomorphic
convex domain is a holomorphically convex domain with respect to the algebra of
bounded holomorphic functions in the domain. The followings are shown in this
paper: In a Riemann domain, a boundedly holomorphic convex domain is a domain
of bounded holomorphy. With some restrictions, the converse is true. The spectrum
of the algebra B of bounded holomorphic functions is an envelope of bounded
holomorphy provided that the completion of B with the topology of uniform
convergence on compact subsets is stable under differentiation. Finally, Stein
manifolds of bounded type are introduced.
