MathDB

Complex coefficient polynomial

Source: China TST 2003

June 29, 2006

algebrapolynomialalgebra unsolvedinequalities

Problem Statement

The

n

roots of a complex coefficient polynomial f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n are

z_1, z_2, \cdots, z_n

. If \sum_{k \equal{} 1}^n |a_k|^2 \leq 1, then prove that \sum_{k \equal{} 1}^n |z_k|^2 \leq n.