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.
Let be a convex function on the interval Then for any we have
Without loss of generality, we assume that
Therefore, there exist such that
On adding (A) and (B),
As is a convex function,
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.