Inequalitites problem! Solve using inequalities only!?

Question

Prove that:

[1/(1 - x²)] + [1/(1 - y²)] ≥ 2/(1 - xy)

given that x and y are positive real numbers using only inequalities.

Answer 1

not true...
if x = 1/3 , y = 2....

left side:
(1/(1-1/9)) + (1/(1-4)) = 9/8 - 1/3 = 19/24

right side:
2/(1-2/3) = 6

the inequality does not hold...

it does not even work if x=2 & y=3...


but i noticed that i will be true if either
.... both x & y are positives less than 1
or
... both are positive but one is less than 1, the other is greater than 1 and the product is less than 1.


§
to aid in the illustration:
[1/(1 - x²)] + [1/(1 - y²)] ??? 2/(1 - xy)
[1/(1 - x²)] + [1/(1 - y²)] - 2/(1 - xy) ??? 0

[ (1 - xy)(1 - y²) + (1 - xy)(1 - x²) - 2(1 - y²)(1 - x²) ] / [(1 - x²)(1 - y²)(1 - xy)] ??? 0

[1 -xy - y² + xy³ + 1 - xy - x² + x³y - 2 + 2y² + 2x² - 2x²y²] / [(1 - x²)(1 - y²)(1 - xy)] ??? 0

consider first the numerator...
(x² + y² - 2xy) + xy(x² + y² - 2xy) which is always positive because xy is positive and the other factor is (x-y)².

now, the denominator...
this is where the problem lies... there are 3 factors... and these can change sign... thus the conclusion... the factors must have a positive product for the inequality to hold.

Answer 2

my brain hurtz really bad!!!