Norm estimates for unitarizable highest weight modules

We consider families of unitarizable highest weight modules (H-lambda)(lamb da is an element of L) on a halfline L. All these modules can be realized a s vector valued holomorphic functions on a bounded symmetric domain D, and the polynomial functions form a dense subset of each module H-lambda, lambd a is an element of L. In this paper we compare the norm of a fixed polynomi al in two Hilbert spaces corresponding to two different parameters. As an a pplication we obtain that for all lambda is an element of L the module of h yperfunction vectors H-lambda(-infinity) can be realized as the space of al l holomorphic functions on D.