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
ahh this is of course if R is very big, so its bigger than 1, so the inequality comes into play from the start!
Thanks very much
but i was worried about one little thing, how do i know that i can talk about
? What if this function is not bounded on the circle?