Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 75941, 15 pages
doi:10.1155/JIA/2006/75941

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

Abstract

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.