A Double-Weighted Refinement of Jensen's Inequality with Applications

摘要

Let (X,,μ) and (Y,,) be two probability measure spaces, I an interval of the real line, fL1(μ), f(x)I for each xX, and a real-valued convex function on I. We show that, if 0 and 1 are two appropriate weight functions on X×Y, then 4(∫xfdμ)≤∫_YA(4;F_0(y)dλ(y)≤∫_x(40f)dμ)where A denotes the arithmetic mean of over the closed interval with end points F0(y) and F1(y), and for -almost all yY's F_k(y) ∫_xf(x)wk(x,y) dμ(x) (k=0,1)Finally, we give nice applications in refining information inequality and some important inequalities between means.