Popoviciu's inequality
Introduction In convex analysis, '''Popoviciu's inequality''' is an inequality about convex functions. It is similar to Jensen's inequality and was found in 1965 by Tiberiu Popoviciu, a Romanian mathematician.
Formulation Let ''f'' be a function from an interval  to . If ''f'' is convex, then for any three points ''x'', ''y'', ''z'' in ''I'',:If a function ''f'' is continuous, then it is convex if and only if the above inequality holds for all ''x'', ''y'', ''z'' from . When ''f'' is strictly convex, the inequality is strict except for ''x'' = ''y'' = ''z''.
Generalizations It can be generalized to any finite number ''n'' of points instead of 3, taken on the right-hand side ''k'' at a time instead of 2 at a time:Let ''f'' be a continuous function from an interval  to . Then ''f'' is convex if and only if, for any integers ''n'' and ''k'' where ''n'' ≥ 3 and , and any ''n'' points  from ''I'',:
Weighted inequality Popoviciu's inequality can also be generalized to a weighted inequality.Let ''f'' be a continuous function from an interval  to . Let  be three points from , and let  be three nonnegative reals such that  and . Then,:
Notes