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$P(x)=a_0+a_1x+...+a_nx^n\in\mathbb{C}[x]$

Source: 4-th Taiwanese Mathematical Olympiad 1995

January 16, 2007

inequalitiesalgebrapolynomialalgebra proposed

Problem Statement

Let

P(x)=a_{0}+a_{1}x+...+a_{n}x^{n}\in\mathbb{C}[x]

, where

a_{n}=1

. The roots of

P(x)

are

b_{1},b_{2},...,b_{n}

, where

|b_{1}|,|b_{2}|,...,|b_{j}|>1

and

|b_{j+1}|,...,|b_{n}|\leq 1

. Prove that

\prod_{i=1}^{j}|b_{i}|\leq\sqrt{|a_{0}|^{2}+|a_{1}|^{2}+...+|a_{n}|^{2}}

.

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