1. ## Taylor series problem

I need some help with the following

Let $\displaystyle f(z)=\sum_{n=0}^{\infty}a_nz^n$ analytic for $\displaystyle |z|<R$ and $\displaystyle |f(z)|\leq M$ for all $\displaystyle |z|<R$.
Show that for every $\displaystyle |z|<\frac {R|a_0|}{M+|a_0|}$, $\displaystyle f(z)\neq 0$.

I got stuck by trying the following:

By Cacuchy's inequality we get that for $\displaystyle r<R$, $\displaystyle \forall n\in N, |a_n|\leq \frac{M}{r^n}$. I need some help on how to take this further in order to show what's required.

2. Just a suggestion. Let $\displaystyle g(z) = f(z)-a_0$. Then $\displaystyle |g(z)|\leqslant M+|a_0|$ whenever $\displaystyle |z|<R$, and $\displaystyle g(0)=0$. I seem to remember that if an analytic function vanishes at the origin and is bounded in a given disc, then there are results limiting how large it can be in a smaller disc (both discs centred at the origin). If you can find a disc within which $\displaystyle |g(z)|<|a_0|$ then clearly $\displaystyle f(z)$ cannot vanish inside that disc.

3. Originally Posted by Opalg
Just a suggestion. Let $\displaystyle g(z) = f(z)-a_0$. Then $\displaystyle |g(z)|\leqslant M+|a_0|$ whenever $\displaystyle |z|<R$, and $\displaystyle g(0)=0$. I seem to remember that if an analytic function vanishes at the origin and is bounded in a given disc, then there are results limiting how large it can be in a smaller disc (both discs centred at the origin). If you can find a disc within which $\displaystyle |g(z)|<|a_0|$ then clearly $\displaystyle f(z)$ cannot vanish inside that disc.
OK, i've been scratching my head for hours and digging through the books, but can't find a result similar to what you mentioned. Can you please eleborate a bit.

Thanks,

SK

4. Originally Posted by skyking
OK, i've been scratching my head for hours and digging through the books, but can't find a result similar to what you mentioned. Can you please eleborate a bit.

Thanks,

SK