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Math Help - Taylor series problem

  1. #1
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    Taylor series problem

    I need some help with the following

    Let f(z)=\sum_{n=0}^{\infty}a_nz^n analytic for |z|<R and |f(z)|\leq M for all |z|<R.
    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<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.
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  2. #2
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    Opalg's Avatar
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    Just a suggestion. Let g(z) = f(z)-a_0. Then |g(z)|\leqslant M+|a_0| whenever |z|<R, 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.
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  3. #3
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    Quote Originally Posted by Opalg View Post
    Just a suggestion. Let g(z) = f(z)-a_0. Then |g(z)|\leqslant M+|a_0| whenever |z|<R, 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
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  4. #4
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    Quote Originally Posted by skyking View Post
    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
    This lemma may be helpful

    Schwarz lemma - Wikipedia, the free encyclopedia
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