Abstract
Suppose that f(x) is strictly increasing, strictly concave, and twice continuously differentiable on a nonempty interval I⊆ℝ, and f′(x) is strictly convex on I. Suppose that xk∈[a,b]⊆I, where 0<a<b, and pk≥0 for k=1,⋯,n, and suppose that ∑k=1npk=1. Let x̄=∑k=1npkxk, and σ2=∑k=1npk(xk−x̄)2. We show ∑k=1npkf(xk)≤f(x̄−θ1σ2), ∑k=1npkf(xk)≥f(x̄−θ2σ2), for suitably chosen θ1 and θ2. These results can be viewed as a refinement of the Jensen's inequality for
the class of functions specified above. Or they can be viewed as a generalization of a refined arithmetic
mean-geometric mean inequality introduced by Cartwright and Field in 1978. The strength of the above result is in bringing the
variations of the xk's into consideration, through σ2.