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.