MathDB
Inequality for three sequences a_i, b_i and c_i - [ILL 1977]

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January 11, 2011
inequalitiesinequalities proposed

Problem Statement

Let mj>0m_j > 0 for j=1,2,…,nj = 1, 2,\ldots, n and a1≤⋯≤an<b1≤⋯≤bn<c1≤⋯≤cna_1 \leq \cdots \leq a_n < b_1 \leq \cdots \leq b_n < c_1 \leq \cdots \leq c_n be real numbers. Prove that (∑j=1nmj(aj+bj+cj))2>3(∑j=1nmj)(∑j=1nmj(ajbj+bjcj+cjaj)).\Biggl( \sum_{j=1}^{n} m_j(a_j+b_j+c_j) \Biggr)^2 > 3 \Biggl( \sum_{j=1}^{n} m_j \Biggr) \Biggl( \sum_{j=1}^{n} m_j(a_jb_j+b_jc_j+c_ja_j) \Biggr).
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