  The main goal of this article is to find the exact difference between a
convex function and its secant, as a limit of positive quantities. This idea
will be expressed as a convex inequality that leads to refinements and
reversals of well established inequalities treating different means. The
significance of these inequalities is to write one inequality that brings
together and refine almost all known inequalities treating the arithmetic,
geometric, harmonic and Heinz means, for numbers and operators.
