The problem:

Let be a polynomial of degree greater than 1, with all of it's zeroes inside the circle . show that .

My idea for a proof:

Since every polynomial over can be writen as , where are all the zeroes of (not necessarily unique). Than we can write .

Now we can say that for every we can find a circle such that

and therfore is analytic in so we get , but since we have and that shows what we needed to show.

I would appreciate some thoughts on this proof as I am not 100% sure it is valid.