A.1 Jensen's Inequality

A function g(x) defined on an open interval I is convex if

for x, y in I and 0 ? p ? 1. Suppose that g(x) has a continuous, nondecreasing derivative g ?(x) on I. The inequality is valid if p = 0 or 1. If x ? y and 0 ? p < 1,

Simplifying this result, we obtain (A-1). If y ? x, a similar analysis again yields (A-1). Thus, if g(x) has a continuous, nondecreasing derivative on I, it is convex.

Lemma. If g(x) is a convex function on the open interval I, then

for all y, x in I, where g ^?(x) is the left derivative of g(x).

Proof: If y ? x ? z > 0, then substituting p =1 ?z/(y ?x) into (A-1) gives

which yields

If v > 0 and z > 0m then (A-1) implies that

which yields

Inequality (A-3) indicates that the ratio [g(y) ?g(x)]/(y ? x) decreases monotonically as y ? x from above and (A-4) implies that this ratio has a lower bound. Therefore, the right derivative g ⁺(x) exists on I. If x ? y ? v > 0,then (A-1) with p = 1 ? v/(x - y)