This is an announcement for the paper "On the best constants in some non-commutative martingale inequalities" by Marius Junge and Quanhua Xu. Abstract: We determine the optimal orders for the best constants in the non-commutative Burkholder-Gundy, Doob and Stein inequalities obtained recently in the non-commutative martingale theory. Archive classification: Operator Algebras, Functional Analysis; Probability Mathematics Subject Classification: 46L53, 46L51 Citation: Bull. London Math. Soc. 37:243--253, 2005 The source file(s), constant.revised.tex: 33956 bytes, is(are) stored in gzipped form as 0505309.gz with size 11kb. The corresponding postcript file has gzipped size 48kb. Submitted from: qx@math.univ-fcomte.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.OA/0505309 or http://arXiv.org/abs/math.OA/0505309 or by email in unzipped form by transmitting an empty message with subject line uget 0505309 or in gzipped form by using subject line get 0505309 to: math@arXiv.org.