I tried to prove it directly (that is, without using some other well-known inequality) by induction and already the first step was harder than what I expected. This first step is: if 0 < a,b <= 1/2, then
sqrt(ab)/sqrt((1 - a)(1 - b)) <= (a + b)/(2 - a + b)
What I did was to square both sides and then what I needed to prove was that
(a + b)^2/(2 - a - b)^2 - ab/((1 - a)(1 - b)) >= 0.
So, I turned the LHS of this expression into a rational expression, whose denominator is clearly >= 0 and whose numerator is
a^2 - a^3 + b^2 - b^3 + a^2b + ab^2 - 2ab. (*)
Since the original inequality is actually an equality when a = b, this is also true for (*) and so I divided (*) by a - b, getting
a - a^2 - b + b^2 = a - b - (a^2 - b^2) = (a - b)(1 - a - b).
So, (*) = (a - b)^2(1 - a - b), which is clearly >= 0.
I have the feeling that what I did was too complicated. Can anyone see a shorter way of proving this (again, avoiding the use of well-known inequalities)?