Hello everyone, I was wondering if anyone could give me a hand with this question.

Let f be an entire function satisfying for , where K is a positive constant. Show using Cauchy's inequality, that for and deduce that f is a quadratic polynomial.

Ok f is entire so is defined and analytic on all of the complex plane. So it is analytic on D(0,R), some big disc.

Then by Taylor's theorem, so has a power series expansion for all z in C, and we can use Cauchy's integral formula for derivatives to get

by the estimation lemma. I understand it all up until this point.

However , nowthe solution I have says that

which I think tells me that

(*) but I have no idea where this comes from! Does anyone have an idea how to help?

For reference Cauchy's inequality in my notes says if f analytic on an open disc and f is bounded on the circle with for some M on , then for ,

If anyone could help me how to use this to get (*) it would be much appreciated