Some Refinements of Discrete Jensen’s Inequality and Some of Its Applications

Jensen’s inequality is sometimes called the king of inequalities [4] because it implies at once the main part of the other classical inequalities (e.g. those by Hölder, Minkowski, Young, and the AGM inequality, etc.). Therefore, it is worth studying it thoroughly and refine it from different points of view. There are numerous refinements of Jensen’s inequality, see e.g. [3-5] and the references in them. In this paper, introducing suitable weight functions, first we give some refinements of discrete Jensen’s inequality, and then using these refinements, we give several important applications in various abstract spaces, which extends the results obtained recently [5,6]. Throughout this paper, we suppose that C is a convex subset of a real vector space, x1, · · · , xn ∈ C, and φ : C → R a convex mapping. Also, we suppose that μ = (μ1, · · · , μm) and λ = (λ1, · · · , λn) are two probability measures; i.e. μi, λj ≥ 0 (1 ≤ i ≤ m, 1 ≤ j ≤ n) with