Geodesic
On Inequalities Complementary to Jensen's
Canadian journal of mathematics, Tome 35 (1983) no. 2, pp. 324-338

Voir la notice de l'article provenant de la source Cambridge University Press

In a paper published in 1975 [1, § 3], D. S. Mitrinovič and P. M. Vasičused the so-called “centroid method” to obtain two new inequalities which are complementary to (the discrete version of) Jensen's inequality for convex functions. In this paper we shall present a very general version of such inequalities using the same geometric ideas used in [1] but not using the centroid method itself. At the same time we shall extend the domain of the inequalities (even in the discrete case), and clarify the value of the constant (λ) appearing in the inequality. We give applications of the theorems to some general means and also to the classical means. Our first result is given asTHEOREM 1. Let v be a nonnegative measure on a σ-algebra of subsets of a set D and let q, f be real v-measurable functions on D such that q(x) > 0, –∞ < x1 ≦ f(x) ≦ x2 < ∞ for all x ∊ D and ∫D qdv = 1. 
Beesack, Paul R. On Inequalities Complementary to Jensen's. Canadian journal of mathematics, Tome 35 (1983) no. 2, pp. 324-338. doi: 10.4153/CJM-1983-018-5
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