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Proof

Using the well-known identity

Then the inequality above equivalent to

Let

The inequality transforms into

Which is clear true. Equality occurs for

Solution

We have

Let then we get

Sitting yields

From inequality we have

and

then we get

Therefore the proof is completed. Equality occurs for

Solution

We have

Because then we have

Thus, it suffices to show that

Letting yields

From inequality we have

So we need to prove that

Now we will show that for are positive real numbers then we have

; and ;

without loss of generality, suppose that . Sitting and then:

which is obviously true. Equality occurs for

is true.

Therefore the proof is completed. Equality occurs for

Proof

Let then we get:

From inequality we have

Let

We have

Therefore we get

Because so from we get

Now we will prove that if be positive real numbers then

Let then we just need to prove that with then

From inequality we have

Therefore we get

Thus, it suffices to show that:

Which is just the third degree inequality

For

Therefore we get

From inequality we have

Therefore the proof is completed. Equality occurs for