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Re: Ky Fan inequality
Posted:
Oct 25, 2012 5:02 AM
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On Tue, 23 Oct 2012, José Carlos Santos wrote:
> Yes, but that's because I should have written:
>
> sqrt(ab)/sqrt((1 - a)(1 - b)) <= (a + b)/(2 - (a + b))
for all a,b with 0 < a,b < 1/2.
> So, I *still* want to prove that
>
> (a + b)^2/(2 - a - b)^2 - ab/((1 - a)(1 - b)) >= 0.
(1-a + 1-b)^2 / (1 - a)(1 - b) <= (a + b)^2 / ab
(1 - a)/(1 - b) + 2 + (1 - b)/(1 - a) <= a/b + 2 + b/a
(1 - a)/(1 - b) + (1 - b)/(1 - a) <= a/b + b/a
(1 - a)/(1 - b) - a/b <= b/a - (1 - b)/(1 - a)
b - ab - a + ab b - ab - a + ab
(b - a)/b(1 - b) <= (b - a)/a(1 - a)
If a < b:
1/b(1 - b) <= 1/a(1 - a)
a(1 - a) <= b(1 - b)
Let f(x) = x(1 - x); f'(x) = 1 - 2x.
For x in (0,1/2), f'(x) > 0; f is increasing over (0,1/2).
If a = b
If b <= a
1/a(1 - a) <= 1/b(1 - b)
b(1 - b) <= a(1 - a)
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