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China Mathematical Olympiad 1994 problem4

Source: China Mathematical Olympiad 1994 problem4

September 17, 2013
algebrapolynomialalgebra unsolved

Problem Statement

Let f(z)=c0zn+c1zn−1+c2zn−2+⋯+cn−1z+cnf(z)=c_0z^n+c_1z^{n-1}+ c_2z^{n-2}+\cdots +c_{n-1}z+c_n be a polynomial with complex coefficients. Prove that there exists a complex number z0z_0 such that ∣f(z0)∣≥∣c0∣+∣cn∣|f(z_0)|\ge |c_0|+|c_n|, where ∣z0∣≤1|z_0|\le 1.
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