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**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.