A refinement of Jensen's inequality is presented. An extra term makes
the inequality tighter when the convex function is ``superquadratic," a
strong
convexity-type condition introduced here. This condition is shown to be
necessary and
sufficient for the refined inequality. It is also shown to be strictly
intermediate
between two points of the scale of convexity. The
refined Jensen's inequality is used to prove a Minkowski inequality
with upper and lower estimates.