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

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 $\displaystyle 1^0=1$.

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.

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?

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

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

Quote:

Originally Posted by

**hedi** Yes,see attached.

Thanks man!