Suppose that $0 < a_0 \le a_1 \le \dots \le a_n$. Prove that the equation $P(z) = a_0z^n + a_1z^{n-1} + \dots + a_{n-1}z + a_n = 0$ has no root in the circle $|z| < 1$.
Prove the equation has no root in the circle |z| < 1
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complex-analysis
complex-numbers
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0|z|<1 is not a circle – 2013-05-07
1 Answers
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We want to show that $a_0z^n+\cdots+a_{n-1}z\neq -a_n$ for $|z|<1$. In fact, by induction we can prove something stronger: that $|a_0z^n+\cdots+a_{n-1}z|
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0I think | a_0z^{n-1} + \cdots + a_{n-1} | < a_{n-1} might be wrong – 2012-09-08