Reference: Nordic Mathematical Contest, 1987-2009, R. Todev.
Question:
Let a, b, and c be real numbers different from 0 and . Prove that inequality
holds. When does the equality hold?
Proof:
We know that a, b and c are real, distinct and also non-zero and also that .
Hence, , we have
, or
On simplifying this, we immediately have
.
A sufficient condition for equality is . If
, then
. which makes the proved inequality a strict one. So,
is a necessary condition for equality too.
-Nalin Pithwa.