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

DOI: 10.1155/JIA/2006/75941

Received: 3 March 2006

Accepted: 12 September 2006

Published: 5 November 2006

Abstract

We consider continuously differentiable means, say https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F75941/MediaObjects/13660_2006_Article_1635_IEq1_HTML.gif -means. As for quasi-arithmetic means https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F75941/MediaObjects/13660_2006_Article_1635_IEq2_HTML.gif , we need an assumption that https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F75941/MediaObjects/13660_2006_Article_1635_IEq3_HTML.gif has no stationary points so that https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F75941/MediaObjects/13660_2006_Article_1635_IEq4_HTML.gif might be continuously differentiable. Introducing quasi-weights for https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F75941/MediaObjects/13660_2006_Article_1635_IEq5_HTML.gif -means would give a satisfactory explanation for the necessity of this assumption. As a typical example of a class of https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F75941/MediaObjects/13660_2006_Article_1635_IEq6_HTML.gif -means, we observe that a skew power mean https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F75941/MediaObjects/13660_2006_Article_1635_IEq7_HTML.gif is a composition of power means if https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F75941/MediaObjects/13660_2006_Article_1635_IEq8_HTML.gif is an integer.

Authors’ Affiliations

(1)
Information Science Division, Department of Arts and Sciences, Osaka Kyoiku University
(2)
Department of Mathematics, Osaka Kyoiku University
(3)
Group of Applied Mathematics and Physics, Department of Basic Technology, Yamagata University
(4)
Department of Mathematics, Shinshu University

Copyright

© 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.