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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 836804, 15 pages
http://dx.doi.org/10.1155/2012/836804
Research Article

A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

1Department of Mathematics Education, Chungbuk National University, Cheongju 361-763, Republic of Korea
2Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

Received 6 November 2012; Accepted 14 December 2012

Academic Editor: Gue Lee

Copyright © 2012 Sangho Kum and Yongdo Lim. 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.

Abstract

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matrices and . Moreover, an interesting generalization of the geometric mean of and to convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.