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 2 of 3: Rewrite both sides
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In plain words

Rewrite both sides

L=∑r=1n−1r(n−r)drL=\sum_{r=1}^{n-1}r(n-r)d_r
Detailed analysis

Let L=∑r=1n−1r(n−r)drL=\sum_{r=1}^{n-1}r(n-r)d_r and Q=∑1≤i<j≤n(∑r=ij−1dr)2Q=\sum_{1\le i<j\le n}(\sum_{r=i}^{j-1}d_r)^2. Each gap drd_r occurs in exactly r(n−r)r(n-r) of the intervals, so the absolute-value double sum is 2L2L. Pairing (i,j)(i,j) with (j,i)(j,i), the squared-difference double sum is 2Q2Q. Hence the original inequality is equivalent to L2≤(n2−1)Q/3L^2\le(n^2-1)Q/3.