where equality holds for . The inequality is sometimes also called Lagrange's
inequality (Mitrinović 1970, p. 42), and can be written in vector form
as
(2)
In two-dimensions, it becomes
(3)
It can be proven by writing
(4)
If
is a constant ,
then .
If it is not a constant, then all terms cannot simultaneously vanish for real , so the solution is complex
and can be found using the quadratic equation
(5)
In order for this to be complex, it must be true
that
(6)
with equality when
is a constant. The vector derivation is much simpler,