consider the function

where is an analytic function on the space (open unit disk in )

can we show that is in fact surjective?

In the bigger picture I would like to thow that , the Banach algebra whose elements are analytic functions on with pointwise defined operations is contained in the space of all bounded analytic function on

The plan is as follows:

Since any is analytic it will also be continuous. is compact, so if I can show that theses functions are in fact surjective then I can use preservation of compactness which implies the image will be compact, and hence closed and bounded.