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China 2010 quiz3 problem 2

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September 11, 2010
algebrapolynomialinequalitiesfloor functiontriangle inequalityinequalities unsolved

Problem Statement

Given positive integer nn, find the largest real number λ=λ(n)\lambda=\lambda(n), such that for any degree nn polynomial with complex coefficients f(x)=anxn+an−1xn−1+⋯+a0f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0, and any permutation x0,x1,⋯ ,xnx_0,x_1,\cdots,x_n of 0,1,⋯ ,n0,1,\cdots,n, the following inequality holds ∑k=0n∣f(xk)−f(xk+1)∣≥λ∣an∣\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|, where xn+1=x0x_{n+1}=x_0.
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