Convex set-valued functions on (0, oo) and their conjugate

Abstract

Let (Ω,Σ,μ) be a σ-finite space and Υ be a Banach space. It is shown that if F : (0,oo) → cl(Y) is a convex continuous set-valued function, then ∫[Ω]y{F°x/y)dμ ⊂ ∫[Ω]ydμF(∫[Ω]xdμ/∫[Ω]ydμ) for all positive p-integrable functions x,y : Ω → R. Moreover, F is convex if and only if its conjugate F[*], F[*](x) = xF(x[-1]), is convex.