Let be an analytic function in the open disk which satisfies for some and for all .

1)Prove that has a zero in or order .

2)Prove that has a removable singularity at .

3)Assume that is entire and that for some and for all . Show that with .

Attempt:1) Let . Then we have . Since , it follows that . Does this also mean that ? If so, then I think I've proven part 1).

Part 2): I think they want me to prove that where .

Using the fact that , I get that which tend to 0 as z tends to 0 because of the sandwich theorem. Proved. (I don't think my proof is valid).

3)I'm not sure.