Does the inequality $\frac{1-x_1x_2}{1-y_1y_2} \leq \frac{1-x_1}{1-y_1} + \frac{1-x_2}{1-y_2}$ hold?

Question

Assume $0<y_i\leq x_i <1 $. where $i=1,2$. Does the below inequality hold?

$$\frac{1-x_1x_2}{1-y_1y_2} \leq \frac{1-x_1}{1-y_1} + \frac{1-x_2}{1-y_2}$$

Answer 1

Note that because $y_1< 1$: $$ 1-y_1\leq 1-y_1y_2\implies \frac{1}{1-y_1}\geq \frac{1}{1-y_1y_2}. $$ Using this inequality for $y_1$ and $y_2$, we can simplify the original inequality to the following: $$ 1-x_1x_2\leq 1-x_1+1-x_2 $$ which is equivalent to: $$ (1-x_1)(1-x_2)\geq 0. $$ The last inequality is true for $x_1,x_2<1$.