A Refinement of Jensen’s Inequality for a Class of Increasing and Concave Functions

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 ∑n k 1pk 1. Let x ∑n k 1pkxk, and σ 2 ∑n k 1pk xk − x . We show ∑n k 1pkf xk ≤ f x−θ1σ 2 , ∑n k 1pkf xk ≥ f x−θ 2σ 2 , 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 meangeometric 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 σ 2.