I don't get what you mean in the first paragraph, but this is (I think) false. It is a standard (although not that well known apparently) result that any open set in has a function defined on it that cannot be extended at all (Domains of Holomorphy). The proof for this (see Remmert, Classical topics in Complex function theory) builds a function that has a sequence accumulating everywhere in the boundary with .

Also is open so it's not compact, which is the problem.

Edit: Mybe this was overkill (although it proves that this inclusion is not valid for any open set). For your case take and note that is a pole.

Edit2: Ha, even in the general case it's an overkill. Just take a point in the boundary and define .