MathLabs

Problem 5

Let nn be a positive integer and let x1,x2,…,xnx_1,x_2,\ldots,x_n be real numbers. Prove that (∑i=1n∑j=1n∣xi−xj∣)2≤2(n2−1)3∑i=1n∑j=1n(xi−xj)2\left(\sum_{i=1}^n\sum_{j=1}^n|x_i-x_j|\right)^2\le\frac{2(n^2-1)}3\sum_{i=1}^n\sum_{j=1}^n(x_i-x_j)^2, with equality if and only if x1,x2,…,xnx_1,x_2,\ldots,x_n form an arithmetic sequence.
Step 1 of 3: Order and take gaps
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In plain words

Order and take gaps

di=xi+1−xi≥0d_i=x_{i+1}-x_i\ge0
Detailed analysis

Because the two double sums are symmetric in the xix_i, relabel so x1≤⋯≤xnx_1\le\cdots\le x_n and set di=xi+1−xi≥0d_i=x_{i+1}-x_i\ge0. For i<ji<j, we then have ∣xj−xi∣=∑r=ij−1dr|x_j-x_i|=\sum_{r=i}^{j-1}d_r and xj−xi=∑r=ij−1drx_j-x_i=\sum_{r=i}^{j-1}d_r.