If are real numbers such that ;
prove that
Lemma: for any
Proof: after simplifying the inequality becomes: for a fixed the quadratic function is convex. hence it attains its maximum at end
points, i.e. or now because also: because
now we may assume that let then:
since by Chebyshev's inequality we have: thus:
since we get: by the Lemma. this proves the upper bound.
proving the lower bound is much easier: since applying the second part of Chebyshev's inequality to and this fact that whenever gives us: