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Abstract: In this paper, we prove that a Banach space X and its dual space have uniform normal structure if . The García-Falset coefficient is estimated by the -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 -constants for a Banach space. [Copyright &y& Elsevier]