Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 75941, 15 pages

Continuously differentiable means

Jun Ichi Fujii,1 Masatoshi Fujii,2 Takeshi Miura,3 Hiroyuki Takagi,4 and Sin-Ei Takahasi3

1Information Science Division, Department of Arts and Sciences, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582-8582, Japan
2Department of Mathematics, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582-8582, Japan
3Group of Applied Mathematics and Physics, Department of Basic Technology, Yamagata University, Yonezawa 992-8510, Japan
4Department of Mathematics, Shinshu University, Asahi, Matsumoto, Nagano 390-8621, Japan

Received 3 March 2006; Revised 7 September 2006; Accepted 12 September 2006

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.


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.