Proof: Complex entire function bounded by a monomial is a polynomial

[Taedrin]

A little explanation here. My professor assigned a homework question without attempting the problem herself. When we were assigned this problem, we were forbidden to use the notion of a Taylor series in our proof (at least not without proving Taylor's Theorem on our own) as we had not covered that topic yet. As it turned out, when the professor DID attempt to solve the problem, she could not complete the proof without the use of a Taylor series, so she threw the problem out.

Unfortunately, she didn't tell us this until we turned in our homework. Since we were having such a horrible time trying to create a proof, several of the students got together to work on it. After 20 hours, we think we found an acceptable attack on a proof which does not use a Taylor series.

But since our professor threw the problem out, she refused to grade our proof. I just graduated today, so the class is over. But I'm dying to know if what we did is actually valid. I'm particularly concerned with my claim that the difference between any two antiderivatives of the same complex function is a constant.

Thank you for any criticism/corrections that you can offer.

Also, my apologies if the LaTeX is a mess, I don't use it that often and often find it difficult to get it to do what I want it to.

 

[mfb]

It is a bit confusing that you give Cauchy's estimate with a region "A", if the same letter is used for a real variable in the problem statement (and 2 lines below Cauchy's estimate, too).
Apart from that, the proof looks good, I don't see an error.