On a Converse of Jensen’s Discrete Inequality

There are many important inequalities which are particular cases of Jensen’s inequality among which are the weighted A − G − H inequality, Cauchy’s inequality, the Ky Fan and Hölder’s inequalities. One can see that the lower bound zero is of global nature since it does not depend on p, x but only on f and the interval I whereupon f is convex. We give in 1 an upper global bound i.e., depending on f and I only which happens to be better than already existing ones. Namely, we prove that