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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 836804, 15 pages
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.
- T. Ando, Topics on Operator Inequalities, Lecture Notes, Hokkaido University, Sapporo, Japan, 1978.
- R. A. Hauser and Y. Lim, “Self-scaled barriers for irreducible symmetric cones,” SIAM Journal on Optimization, vol. 12, no. 3, pp. 715–723, 2002.
- Yu. E. Nesterov and M. J. Todd, “Self-scaled barriers and interior-point methods for convex programming,” Mathematics of Operations Research, vol. 22, no. 1, pp. 1–42, 1997.
- R. Bhatia, “On the exponential metric increasing property,” Linear Algebra and its Applications, vol. 375, pp. 211–220, 2003.
- R. Bhatia and J. Holbrook, “Riemannian geometry and matrix geometric means,” Linear Algebra and its Applications, vol. 413, no. 2-3, pp. 594–618, 2006.
- M. Moakher, “A differential geometric approach to the geometric mean of symmetric positive-definite matrices,” SIAM Journal on Matrix Analysis and Applications, vol. 26, no. 3, pp. 735–747, 2005.
- M. Moakher, “On the averaging of symmetric positive-definite tensors,” Journal of Elasticity, vol. 82, no. 3, pp. 273–296, 2006.
- C. J. Hillar and C. R. Johnson, “Symmetric word equations in two positive definite letters,” Proceedings of the American Mathematical Society, vol. 132, no. 4, pp. 945–953, 2004.
- C. R. Johnson and C. J. Hillar, “Eigenvalues of words in two positive definite letters,” SIAM Journal on Matrix Analysis and Applications, vol. 23, no. 4, pp. 916–928, 2002.
- J. Lawson and Y. Lim, “Solving symmetric matrix word equations via symmetric space machinery,” Linear Algebra and its Applications, vol. 414, no. 2-3, pp. 560–569, 2006.
- M. Atteia and M. Raïssouli, “Self dual operators on convex functionals; geometric mean and square root of convex functionals,” Journal of Convex Analysis, vol. 8, no. 1, pp. 223–240, 2001.
- J. A. Johnstone, V. R. Koch, and Y. Lucet, “Convexity of the proximal average,” Journal of Optimization Theory and Applications, vol. 148, no. 1, pp. 107–124, 2011.
- H. H. Bauschke, R. Goebel, Y. Lucet, and X. Wang, “The proximal average: basic theory,” SIAM Journal on Optimization, vol. 19, no. 2, pp. 766–785, 2008.
- S. Kim, J. Lawson, and Y. Lim, “The matrix geometric mean of parameterized, weighted arithmetic and harmonic means,” Linear Algebra and its Applications, vol. 435, no. 9, pp. 2114–2131, 2011.
- R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, USA, 1970.
- R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317, Springer, Berlin, Germany, 1998.