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