Most general linear inequality for convex functions of a real variable

We provide necessary and sufficient conditions on the real coefficients in order that the inequality be verified for every convex function f(defined on some real interval I) and all such that (Theorems 2, 3). Our conditions are simple linear inequalities in . Thus we obtain the most general (finite) linear inequality for convex functions of a real variable, which contains as special cases all classical inequalities of this form (given by Jensen, Steffensen, Szegö Bellman, Brunk, Olkin, Petrović, Dragomirescu and Ivan, etc., all providing only sufficient conditions for (I n ) to be satisfied), as well as some new ones. Another noteworthy result (Theorem 4): when these conditions are verified, and f is strictly convex, we provide necessary and sufficient conditions for equality to occur in (I n ). As a by-product we obtain necessary and sufficient conditions on for (I n ) to be verified for every nondecreasing function fand all in I(Theorem 1), as well as the most general linear inequalities for monotone and respectively convex sequences (Theorems 5, 6).