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: CONTINUOUSLY DIFFERENTIABLE MEANS JUN ICHI FUJII, MASATOSHI FUJII, TAKESHI MIURA, | CONTINUOUSLY DIFFERENTIABLE MEANS JUN ICHI FUJII MASATOSHI FUJII TAKESHI MIURA HIROYUKI TAKAGI AND SIN-EI TAKAHASI Received 3 March 2006 Revised 7 September 2006 Accepted 12 September 2006 We consider continuously differentiable means say c1 -means. As for quasi-arithmetic means Qf x1 . xn we need an assumption that f has no stationary points so that Qf might be continuously differentiable. Introducing quasi-weights for c1 -means would give a satisfactory explanation for the necessity of this assumption. As a typical example of a class of c1 -means we observe that a skew power mean Mt is a composition of power means if t is an integer. Copyright 2006 Jun Ichi Fujii et al. 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 Let M x1 . xn be a continuously differentiable -variable positive function on 0 to . Then throughout this paper M is called a continuously differentiable mean or shortly c1 -mean if M satisfies i M is monotone increasing in each term ii M a . a a for all positive numbers a. A mean M is called homogeneous if M satisfies M ax-Ị_ . axn aMfa . xn for all a xk 0. Almost all classical means are homogeneous c1-ones. The Kubo-Ando operator means in 6 and chaotic ones in 2 are c1-means. Here note that numerical Kubo-Ando means Kf a b are defined by J b Kf a b af ffff Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 75941 Pages 1-15 DOI JIA 2006 75941 2 Continuously differentiable means for positive operator monotone functions f which form a special class of numerical means. Let f be a continuously differentiable monotone function on 0 to with no stationary points that is f x 0 for all x 0. In this case f 1 is also continuously differentiable. Let w wk be a weight that is a set of nonnegative numbers wk with y k wk 1. For .
