The following inequality and its solution is dedicated to my beloved teacher, Christos Patilas…
If are positive real numbers, with , prove that
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