help for question on bound of f(0)

• Nov 16th 2012, 03:22 AM
alphabeta89
help for question on bound of f(0)
Let $\displaystyle f(z)$ be a function which is analytic on and inside the circle $\displaystyle |z| = 1$ and which satisfies $\displaystyle |f(z)|\leq{M}$ for all $\displaystyle z$ on the circle $\displaystyle |z| = 1$. Prove that $\displaystyle |f(0)|\leq{M}$.

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

How do i proceed to prove that $\displaystyle |f(0)|\leq{M}$? Any help/suggestion is welcome!:D
• Nov 16th 2012, 04:07 AM
FernandoRevilla
Re: help for question on bound of f(0)
Quote:

Originally Posted by alphabeta89
How do i proceed to prove that $\displaystyle |f(0)|\leq{M}$? Any help/suggestion is welcome!:D

Hint: $\displaystyle 0!=1$ and $\displaystyle 1^0=1$.
• Nov 16th 2012, 04:13 AM
hedi
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.
• Nov 16th 2012, 04:17 AM
alphabeta89
Re: help for question on bound of f(0)
Quote:

Originally Posted by FernandoRevilla
Hint: $\displaystyle 0!=1$ and $\displaystyle 1^0=1$.

Hi, does Cauchy's inequality hold for $\displaystyle |f^{(0)}(0)|$ in this case?
• Nov 16th 2012, 05:04 AM
hedi
Re: help for question on bound of f(0)
Yes,see attached.
• Nov 16th 2012, 05:15 AM
alphabeta89
Re: help for question on bound of f(0)
Quote:

Originally Posted by hedi
Yes,see attached.

Thanks man!