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International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 9, Pages 527-532
http://dx.doi.org/10.1155/S0161171203206116

On vector-valued Hardy martingales and a generalized Jensen's inequality

Department of Mathematics and Physics, University of Louisiana at Monroe, Monroe, LA 71209, USA

Received 24 June 2002

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.