# Math Help - Taylor series problem

1. ## Taylor series problem

I need some help with the following

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

I got stuck by trying the following:

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

2. Just a suggestion. Let $g(z) = f(z)-a_0$. Then $|g(z)|\leqslant M+|a_0|$ whenever $|z|, and $g(0)=0$. 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 $|g(z)|<|a_0|$ then clearly $f(z)$ cannot vanish inside that disc.

3. Originally Posted by Opalg
Just a suggestion. Let $g(z) = f(z)-a_0$. Then $|g(z)|\leqslant M+|a_0|$ whenever $|z|, and $g(0)=0$. 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 $|g(z)|<|a_0|$ then clearly $f(z)$ 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