In article <aei7kuF620vU1@mid.individual.net>, José Carlos Santos <firstname.lastname@example.org> wrote:
> .... This first step is: if 0 < a,b <= 1/2, > then > > sqrt(ab)/sqrt((1 - a)(1 - b)) <= (a + b)/(2 - a + b) > > .... Can anyone see > a shorter way of proving this (again, avoiding the use of well-known > inequalities)? ....
I assume your (2 - a + b) is a typo for (2 - a - b).
Transform the problem a bit. You want to prove that
(a + b)/sqrt(ab) >= ((1 - a) + (1 - b))/sqrt((1 - a)(1 - b))
If you put each side over the appropriate common denominator, then the numerators come out the same on both sides. Assuming wolog a >= b, it's easy to show that the common numerator is positive, so cancel it. Then the necessary inequality between the denominators is also elementary.