```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,>> 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).Sure. Thanks.> 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.Cute. Thanks.Best regards,Jose Carlos Santos
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