Date: Oct 23, 2012 8:58 AM
Author: Jose Carlos Santos
Subject: Re: Ky Fan inequality
On 23-10-2012 1:33, Ken Pledger wrote:
>> .... This first step is: if 0 < a,b <= 1/2,
>> 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
Jose Carlos Santos