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Geometry and inequalities of geometric mean

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Abstract

We study some geometric properties associated with the t-geometric means A ♯ t B:= A 1/2(A −1/2 BA −1/2)t A 1/2 of two n × n positive definite matrices A and B. Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding m pairs of positive definite matrices is posted.

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Authors and Affiliations

  1. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, 19 Nguyen Huu Tho Street, Tan Phong Ward, District 7, Ho Chi Minh City, Vietnam

    Trung Hoa Dinh

  2. Faculty of Civil Engineering, Ton Duc Thang University, 19 Nguyen Huu Tho Street, Tan PhongWard, District 7, Ho Chi Minh City, Vietnam

    Trung Hoa Dinh

  3. Department of Mathematics and Statistics, Auburn University, Duncan Dr, Auburn, Alabama, 36849, USA

    Sima Ahsani & Tin-Yau Tam

Authors
  1. Trung Hoa Dinh
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  2. Sima Ahsani
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  3. Tin-Yau Tam
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Corresponding author

Correspondence to Trung Hoa Dinh.

Additional information

In memory of Professor Miroslav Fiedler who passed away on November 20, 2015

The research of the first author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant number 101.04-2014.40.

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Dinh, T.H., Ahsani, S. & Tam, TY. Geometry and inequalities of geometric mean. Czech Math J 66, 777–792 (2016). https://doi.org/10.1007/s10587-016-0292-8

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  • DOI: https://doi.org/10.1007/s10587-016-0292-8

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