We establish a generalized Jensen's inequality for analytic vector-valued functions on 𝕋N using a monotonicity
property of vector-valued Hardy martingales. We then discuss how this
result extends to functions on a compact abelian group G, which are
analytic with respect to an order on the dual group. We also give a
generalization of Helson and Lowdenslager's version of Jensen's
inequality to certain operator-valued analytic functions.