On James and von Neumann–Jordan constants and sufficient conditions for the fixed point property

In this paper, we prove that a Banach space X and its dual space X∗ have uniform normal structure if CNJ(X) < (1+√3)/2. The García-Falset coefficient R(X) is estimated by the CNJ(X)-constant and the weak orthogonality coefficient introduced by B. Sims. Finally, we present an affirmative answer to a conjecture by L.Maligranda concerning the relation between the James and CNJ(X)-constants for a Banach space. © 2005 Elsevier Inc. All rights reserved.