Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: THE JAMES CONSTANT OF NORMALIZED NORMS ON R2 | THE JAMES CONSTANT OF NORMALIZED NORMS ON R2 WEERAYUTH NILSRAKOO AND SATIT SAEJUNG Received 28 June 2005 Accepted 13 September 2005 We introduce a new class of normalized norms on R2 which properly contains all absolute normalized norms. We also give a criterion for deciding whether a given norm in this class is uniformly nonsquare. Moreover an estimate for the James constant is presented and the exact value of some certain norms is computed. This gives a partial answer to the question raised by Kato et al. Copyright 2006 W. Nilsrakoo and S. Saejung. 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. 1. Introduction and preliminaries A norm II II on C2 resp. R2 is said to be absolute if ỊỊ z w z w for all z w e C resp. R and normalized if 11 1 0 II 11 0 1 II 1. The Ểp-norms II IIp are such examples ll z w ll p - z p w p 1 p if 1 p TO 11 max 1 z w ifp TO. Let AN2 be the family of all absolute normalized norms on C2 resp. R2 and v2 the family of all continuous convex functions y on 0 1 such that y 0 y 1 1 and max 1 - t t y t 1 0 t 1 . According to Bonsall and Duncan 1 AN2 and v2 are in a one-to-one correspondence under the equation y t 11 1 - t oil 0 t 1 . Indeed for all y e v2 let z M z ll z w lly if z w 0 0 if z w 0 0 . 0 Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 26265 Pages 1-12 DOI JIA 2006 26265 2 The James constant of normalized norms on R2 Then II Hy e AN2 and II by satisfies . From this result we can consider many non-fp-type norms easily. Now let 1 - t p tp 1 p if 1 p TO max 1 - t t if p TO. Then yp t e v2 and as is easily seen the Ểp-norm II lip is associated with yp. If X is a Banach space then X is uniformly nonsquare if there exists s e 0 1 such that for any x y e SX either x yll 2 1 - Ổ or x- yll 2 1 - s where SX x e X x
