Please show that :

if and

then

(Itwasntme)

thanks very much

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- April 29th 2008, 12:20 AMXingyuanAn Intergral Inequlity
Please show that :

if and

then

(Itwasntme)

thanks very much - April 29th 2008, 04:33 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. - April 29th 2008, 04:46 AMPaulRS
- May 2nd 2008, 07:00 PMXingyuan
excellent proof. very powerful. (Evilgrin)

- May 2nd 2008, 07:07 PMPaulRS