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Suppose I have an $N^{\text{th}}$ degree polynomial $P_N(z)=\sum_{i=0}^N a_i z^i$ where $\{a_i\}_{i=0}^N$ are complex numbers such that $|a_N|> \sum_{i=0}^{N-1}|a_i|$, can I claim that all its roots have magnitude lesser than unity? Simulation results in Matlab have ALWAYS given me roots with magnitude lesser than unity. But I lack the complex analysis background needed to prove this.

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Let $z\in \mathbb{C}$ such that $P(z)=a_Nz^N+\cdots+a_0=0$. Suppose that $|z|>1$. Then

$$|a_N||z|^N=|-a_{N-1}z^{N-1}-\cdots -a_0|\leq |a_{N-1}| |z|^{N-1}+\cdots+|a_0|$$ As $|z|^k<|z|^{N}$ for $k=0,\cdots,N-1$, we get $$ |a_N||z|^N \leq (|a_{N-1}|+\cdots+|a_0|)|z|^{N}$$ hence $$ |a_N|\leq (|a_{N-1}|+\cdots+|a_0|)$$ a contradiction with your hypothesis.

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