A convexity inequality

Jensen's inequality for concave functions J J (f) du < j (ff dt) (1) or for convex functions i ( fdI) < J J(f) dA (2) is valid when ,L is a probability measure. In Sections 2 and 7 we extend (1) and (2) to arbitrary positive measures under the assumption that J is positively homogeneous. In Section 6 we show how the H6lder, Minkowski, and Hanner inequalities follow directly from (1) and (2). In particular, we show the one-to-one correspondence between inequalities such as (2) and continuous convex functions j: [0, +oo) -> 1R such that lim,+OO j (t)/t exists and is finite.