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Re: Ky Fan inequality
Posted:
Oct 22, 2012 8:33 PM


In article <aei7kuF620vU1@mid.individual.net>, José Carlos Santos <jcsantos@fc.up.pt> 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 wellknown > 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.
Ken Pledger.



