Function theory in the bfd-norm on an elliptical region

Let E be the open region in the complex plane bounded by an ellipse. The B. and F. Delyon norm ‖⋅‖bfd on the space Hol(E) of holomorphic functions on E is defined by‖f‖bfd=defsupT∈Fbfd(E)⁡‖f(T)‖, where Fbfd(E) is the class of operators T such that the closure of the numerical range of T is contained in E. The name of the norm recognizes a celebrated theorem of the brothers Delyon, which implies that ‖⋅‖bfd is equivalent to the supremum norm ‖⋅‖∞ on Hol(E). The purpose of this paper is to develop the theory of holomorphic functions of bfd-norm less than or equal to one on E. To do so we shall employ a remarkable connection between the bfd norm on Hol(E) and the supremum norm ‖⋅‖∞ on the space H∞(G) of bounded holomorphic functions on the symmetrized bidisc, the domain G in C2 defined byG=def{(z+w,zw):|z|