Let $a_1$, $a_2$, $a_3$ be three different real numbers. Define real numbers $b_1$ $b_2$ $b_3$ as $$b_1 = (1+ \frac{a_1a_2}{a_1-a_2})(1+ \frac{a_1a_3}{a_1-a_3}) $$ $$b_2 = (1+ \frac{a_2a_1}{a_2-a_1})(1+ \frac{a_2a_3}{a_2-a_3}) $$ $$b_3 = (1+ \frac{a_3a_1}{a_3-a_1})(1+ \frac{a_3a_2}{a_3-a_2}) $$ Prove that $$ 1+ |a_1b_1+a_2b_2+a_3b_3|\le (1+|a_1|)(1+|a_2|)(1+|a_3|)$$
I have no idea how to simplify the left side.. but the right side could perhaps be shrunk to $8\sqrt{|a_1a_2a_3|}$?
This question is classified as "variable substitution".. could this help?