# Math Help - help for question on bound of f(0)

1. ## help for question on bound of f(0)

Let $f(z)$ be a function which is analytic on and inside the circle $|z| = 1$ and which satisfies $|f(z)|\leq{M}$ for all $z$ on the circle $|z| = 1$. Prove that $|f(0)|\leq{M}$.

By Cauchy's inequality i managed to obtain
$|f'(0)|\leq{\frac{1!M}{1^1}=M}$.

How do i proceed to prove that $|f(0)|\leq{M}$? Any help/suggestion is welcome!

2. ## Re: help for question on bound of f(0)

Originally Posted by alphabeta89
How do i proceed to prove that $|f(0)|\leq{M}$? Any help/suggestion is welcome!
Hint: $0!=1$ and $1^0=1$.

3. ## Re: help for question on bound of f(0)

use the Cauchy integral formula to evaluate and estimate f(0) as an integral over the unit circle.

4. ## Re: help for question on bound of f(0)

Originally Posted by FernandoRevilla
Hint: $0!=1$ and $1^0=1$.
Hi, does Cauchy's inequality hold for $|f^{(0)}(0)|$ in this case?

5. ## Re: help for question on bound of f(0)

Yes,see attached.

6. ## Re: help for question on bound of f(0)

Originally Posted by hedi
Yes,see attached.
Thanks man!