Advances in Operator Theory

The matrix power means and interpolations

Trung Hoa Dinh, ‎Raluca Dumitru, and Jose A‎. ‎Franco

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


‎It is well-known that the Heron mean is a linear interpolation between the arithmetic and the geometric means while the matrix power mean $P_t(A,B):= A^{1/2}\left(\frac{I+(A^{-1/2}BA^{-1/2})^t}{2}\right)^{1/t}A^{1/2}$ interpolates between the harmonic‎, ‎the geometric‎, ‎and the arithmetic means‎. ‎In this article‎, ‎we establish several comparisons between the matrix power mean‎, ‎the Heron mean‎, ‎and the Heinz mean‎. ‎Therefore‎, ‎we have a deeper understanding about the distribution of these matrix means‎.

Article information

Adv. Oper. Theory, Volume 3, Number 3 (2018), 647-654.

Received: 5 January 2018
Accepted: 28 February 2018
First available in Project Euclid: 4 April 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A63: Operator inequalities
Secondary: 47A64‎ ‎47A56

Kubo-Ando means ‎interpolation‎ ‎arithmetic mean geometric mean harmonic mean ‎Heron means ‎Heinz means ‎power means


Dinh, Trung Hoa; Dumitru, ‎Raluca; ‎Franco, Jose A‎. The matrix power means and interpolations. Adv. Oper. Theory 3 (2018), no. 3, 647--654. doi:10.15352/aot.1801-1288.

Export citation


  • A. Begea, J. Bukor, and J. T. Tóthb, On log-convexity of power mean, Ann. Math. Inform. 42 (2013), 3–7.
  • R. Bhatia, Interpolating the arithmetic-geometric mean inequality and its operator version, Linear Algebra Appl. 413 (2006), no. 2-3, 355–363.
  • R. Bhatia and F. Kittaneh, The matrix arithmetic-geometric mean inequality revisited, Linear Algebra Appl. 428 (2008), no. 8-9, 2177–2191.
  • R. Bhatia, Y. Lim, and T. Yamazaki, Some norm inequalities for matrix means, Linear Algebra Appl. 501 (2016), 112–122.
  • T. H. Dinh, R. Dumitru, and J. A. Franco, On a conjecture of Bhatia, Lim and Yamazaki, Linear Algebra Appl. 532 (2017), 140–145.
  • F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1979/80), no. 3, 205–224.
  • Y. Lim and M. Pálfia, Matrix power means and the Karcher mean, J. Funct. Anal. 262 (2012), no. 4, 1498–1514.
  • L. Matejicka, Short note on convexity of power mean, Tamkang J. Math. 46 (2015), no. 4, 423–426.