Please show that :

if and

then

(Itwasntme)

thanks very much

Printable View

- Apr 28th 2008, 11:20 PMXingyuanAn Intergral Inequlity
Please show that :

if and

then

(Itwasntme)

thanks very much - Apr 29th 2008, 03:33 AMPaulRS
Since we may apply the geometric series

Now I'll show that: valid

Consider the Riemannian sum:

By the Power Mean inequality we have:

Thus: and:

So we have:

This may also be shown applying Hölder's Inequality

Assuming uniform convergence *

Applying the inequality we've just proved:

Which yields:

This last step is correct since:

By the way, there's equality iff the function is constant. - Apr 29th 2008, 03:46 AMPaulRS
- May 2nd 2008, 06:00 PMXingyuan
excellent proof. very powerful. (Evilgrin)

- May 2nd 2008, 06:07 PMPaulRS