ON JENSEN'S INEQUALITY

Let (X, A,μ) and (Y, B, λ) be two probability measore spaces, I an interval of the real line, f Є L1(μ), f(x) Є I for each x Є X, and φ a realvalued convex function on I. We show that, if w0 and w1 are two appropriate weight functions on X*Y, then φ(∫xfdμ) ≤ ∫Y A(φ, F0(y), F1(y)) dλ(y) ≤∫x (φ f) dμ where A denotes the arthmetic mean of φ over the closed interval with end points F0(y), and for λ-almost all yЄY's Fk(y) = ∫x f(x) wk(x,y)dμ (x) (k=0,1). At the end, we give some nice applications refining AGM and the information inequalities.

Keywords:

Jensen's Inequality , Weight function , Mean , Information Inequality