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let $f(z)=z^9+z^5+8z^3+2z+1$, well, $|f(z)-z^9|=|z^5+8z^3+2z+1|<|2|^9\text{ at } |z|=2$ so $f$ has $9$ zeroes inside $|z|<2$

and $|f(z)-8z^3|=|z^9+z^5+2z+1|<5<8|z|^3\text{ at } |z|=1$ so inside $|z|<1$ $f$ has 3 zeroes, so $6$ zeroes are there inside $1, could any one tell me whether I have properly applied Rouché’s theorem to this problem? Thank you for reply.

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    You are correct and have supplied a complete proof. Source: I just finished a course in Comple$x$ Analysis at my university.2012-07-22

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MSE community hereby certifies the proof as correct and complete. (Maybe add a word about no roots on boundary circles).