Hadamard Inequality and a Refinement of Jensen Inequality for Set—Valued Functions

Abstract

We present the following set-valued analogue of the Hadamard inequality: Let Y be a Banach space, I be an open interval and let F: I ↦ cl(Y) be a continuous and convex set-valued function. Then

$${F(a)+F(b)\over 2}\subset{1\over b-a}\int_a^b\ F(x)dx\subset F \bigg({a+b\over 2}\bigg)$$

, for every a, b ∈ I, a < b. Some refinement of Jensen inequality for set-valued functions is also given.