We establish a generalized Jensen's inequality for analytic vector-valued functions on 𝕋 N 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.