>> .... 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)) > > i.e. sqrt(a/b) + sqrt(b/a) >= > sqrt((1 - a)/(1 - b)) + sqrt((1 - b)/(1 - a)) > > i.e. sqrt(a/b) - sqrt((1 - a)/(1 - b)) >= > sqrt((1 - b)/(1 - a)) - sqrt(b/a). > > 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.