MathLabs

Problem 2

Show that the inequality ∑i=1n∑j=1n∣xi−xj∣≤∑i=1n∑j=1n∣xi+xj∣\sum_{i=1}^n\sum_{j=1}^n\sqrt{|x_i-x_j|}\le\sum_{i=1}^n\sum_{j=1}^n\sqrt{|x_i+x_j|} holds for all real numbers x1,x2,…,xnx_1,x_2,\ldots,x_n.
Step 2 of 5: Shift every variable by tt and isolate the right side
Previous stepNext step
F(t)=∑i=1n∑j=1n∣xi+xj+2t∣F(t)=\sum_{i=1}^n\sum_{j=1}^n\sqrt{|x_i+x_j+2t|}
Detailed analysis

Replacing xix_i by xi+tx_i+t for every ii leaves ∑i,j∣xi−xj∣\sum_{i,j}\sqrt{|x_i-x_j|} unchanged, while the right side becomes the function F(t)=∑i=1n∑j=1n∣xi+xj+2t∣F(t)=\sum_{i=1}^n\sum_{j=1}^n\sqrt{|x_i+x_j+2t|}; it suffices to prove the original inequality with the right side replaced by min⁡tF(t)\min_t F(t).