Popoviciu's Inequality
Popoviciu’s inequality will be used in the same manner as Jensen’s inequality. But we must note that this inequality is stronger, i.e. in some cases this inequality can be a powerful tool for proving other inequalities where Jensen’s inequality does not work.
Popoviciu’s Inequality
Let be a convex function on the interval Then for any we have
Without loss of generality, we assume that
If then
Therefore, there exist such that
On adding (A) and (B),
As is a convex function,
and
After adding together the last three inequalities we obtain the required inequality.
The case when is considered similarly, bearing in mind that and
Note: When is a concave function, the inequality gets reversed.