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Open Access

Continuously differentiable means

  • Jun Ichi Fujii1Email author,
  • Masatoshi Fujii2,
  • Takeshi Miura3,
  • Hiroyuki Takagi4 and
  • Sin-Ei Takahasi3
Journal of Inequalities and Applications20062006:75941

Received: 3 March 2006

Accepted: 12 September 2006

Published: 5 November 2006


We consider continuously differentiable means, say -means. As for quasi-arithmetic means , we need an assumption that has no stationary points so that might be continuously differentiable. Introducing quasi-weights for -means would give a satisfactory explanation for the necessity of this assumption. As a typical example of a class of -means, we observe that a skew power mean is a composition of power means if is an integer.

Authors’ Affiliations

Information Science Division, Department of Arts and Sciences, Osaka Kyoiku University, Kashiwara, Osaka, Japan
Department of Mathematics, Osaka Kyoiku University, Kashiwara, Osaka, Japan
Group of Applied Mathematics and Physics, Department of Basic Technology, Yamagata University, Yonezawa, Japan
Department of Mathematics, Shinshu University, Matsumoto, Nagano, Japan


© Jun Ichi Fujii et al. 2006

This article is published under license to BioMed Central Ltd. 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.