*Problem:*

If are non-negative numbers prove that

.

*Solution:*

Lemma: .

Back to the inequality now, multiply both sides by . Then we have that .

But from the lemma we reduce the current inequality to

.

It also holds . Multiplying the last inequality with we get that .

So, it suffices to prove that

or

,

which reduces to the obvious inequality

.

Equality occurs for and also for or any cyclic permutation,* Q.E.D.
*

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