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IMC 2009 Day 1 P4

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July 15, 2014
algebrapolynomialIMCcollege contests

Problem Statement

Let p(z)=a0+a1z+a2z2+⋯+anznp(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n be a complex polynomial. Suppose that 1=c0≥c1≥⋯≥cn≥01=c_0\ge c_1\ge \cdots \ge c_n\ge 0 is a sequence of real numbers which form a convex sequence. (That is 2ck≤ck−1+ck+12c_k\le c_{k-1}+c_{k+1} for every k=1,2,⋯ ,n−1k=1,2,\cdots ,n-1 ) and consider the polynomial q(z)=c0a0+c1a1z+c2a2z2+⋯+cnanzn q(z)=c_0a_0+c_1a_1z+c_2a_2z^2+\cdots +c_na_nz^n Prove that : max⁡∣z∣≤1q(z)≤max⁡∣z∣≤1p(z) \max_{|z|\le 1}q(z)\le \max_{|z|\le 1}p(z)
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