*Problem:*

For all prove that

.

*Solution:*

It holds that . Thus, we only need to prove that

.

Observe that

.

From the above result we have to prove now that

.

Using the AM – GM Inequality we get that

.

So, it is enough to prove that

.

The last Inequality can be reduced to the .

So, it is enough to check the Inequality

.

This one can be prove by the following way. Set and , with and remake the Inequality to the form

.

Then, the Inequality substitutes to

.

But this one holds because we have that

,

due to the AM – GM Inequality and the hypothesis , *Q.E.D.*

**P.S** The following nice Inequality also holds:

.

This Inequality belongs to *Thomas Mildorf* and the proof for this Inequality is the same as the above.