If are positive real numbers satisfying the equality , then prove that
Without loss of generality, assume that .
using Chebyshev’s Inequality we get:
Moreover, From the AM-GM Inequality we have that
Therefore, it suffices to prove that
The above inequality can be written now as
, or .
From the AM-GM Inequality once again, we acquire that
And thus, it is enough to check that
which is equivalent to the obvious inequality