*Problem:*

Let be positive real numbers. Prove that

.

*Solution:*

Let us use Holder’s inequality, that is

.

So, the current inequality reduces to:

.

Now, from the Power-Mean inequality we acquire that

and .

And therefore we have that

.

Let us divide the last fraction by . Then by setting , the last inequality becomes

.

So, we only need to prove that

,

or

, or .

Setting the above inequality becomes .

But the last inequality is equivalent to which is obviously true since

and

, *Q.E.D.*

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