*The following inequality and its solution is dedicated to my beloved teacher, Christos Patilas…*

*Problem:*

If are positive real numbers, with , prove that

.

*Solution:*

Without loss of generality assume that . Then from Chebyshev’s inequality we have

.

*Lemma (Vasile Cirtoaje):*

*If and then it holds that *

.

*Proof of the Lemma:*

We know that . So, it suffices to prove that .

Let us denote by . Substituting them to the above inequality we get that

,

which reduces to the inequality

,

which is obvious since .

Back to our inequality now, from the above lemma we deduce that:

.

For convenience denote by the . Therefore we have that

.

Thus it remains to prove that

.

But . So, the last fraction is of the form .

After that we get

.

Conclusion follows from the obvious inequality

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