Let be positive real numbers such that . Prove that
We divide the inequality with . Then we get that . Let .
We need to prove that
But from the hypothesis we get that .
From Jensen’s inequality we get that
After some calculations, we get the desired result, that is
From AM-GM inequality we get that
Similarly we can prove that
So we prooved that . Similarly we can prove that . Multiplying these inequalities the desired inequality follows, Q.E.D.