## Annals of Functional Analysis

### Some matrix inequalities for weighted power mean

Maryam Khosravi

#### Abstract

In this paper, we prove that, for any positive definite matrices $A,B$, and real numbers $\nu,\mu,p$ with $-1\leq p\lt 1$ and $0\lt \nu\leq\mu\lt 1$, we have

$$\frac{\nu}{\mu}(A\nabla_{\mu}B-A\sharp_{p,\mu}B)\leq{A\nabla_{\nu}B-A\sharp_{p,\nu}B}\leq\frac{1-\nu}{1-\mu}(A\nabla_{\mu}B-A\sharp_{p,\mu}B),$$ where $\nabla_{\nu}$ and $\sharp_{p,\nu}$ stand for weighted arithmetic and power mean, respectively. In the special cases when $p=0,1$, this inequality can be considered as a generalization of harmonic-arithmetic and geometric-arithmetic means inequalities and their reverses.

Applying this inequality, some inequalities for the Heinz mean and determinant inequalities related to weighted power means are obtained.

#### Article information

Source
Ann. Funct. Anal., Volume 7, Number 2 (2016), 348-357.

Dates
Accepted: 18 October 2015
First available in Project Euclid: 8 April 2016

https://projecteuclid.org/euclid.afa/1460141560

Digital Object Identifier
doi:10.1215/20088752-3544480

Mathematical Reviews number (MathSciNet)
MR3484388

Zentralblatt MATH identifier
1337.15020

#### Citation

Khosravi, Maryam. Some matrix inequalities for weighted power mean. Ann. Funct. Anal. 7 (2016), no. 2, 348--357. doi:10.1215/20088752-3544480. https://projecteuclid.org/euclid.afa/1460141560

#### References

• [1] H. Alzer, C. M. da Fonseca, and A. Kovačec, Young-type inequalities and their matrix analogues, Linear Multilinear Algebra. 63 (2015), no. 3, 622–635.
• [2] S. Furuichi and N. Minculete, Alternative reverse inequalities for Young’s inequality, J. Math. Inequal. 5 (2011), no. 4, 595–600.
• [3] R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge Univ. Press, Cambridge, 203.
• [4] F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl. 361 (2010), no. 1, 262–269.
• [5] F. Kittaneh and Y. Manasrah, Reverse Young and Heinz inequalities for matrices, Linear Multilinear Algebra 59 (2011), no. 9, 1031–1037.
• [6] M. Krnić, N. Lovričević, and J. Pečarić, Jensen’s operator and applications to mean inequalities for operators in Hilbert space, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), no. 1, 1–14.
• [7] W. Liao and J. Wu, Matrix inequalities for the difference between arithmetic mean and harmonic mean, Ann. Funct. Anal. 6 (2015), no. 3, 191–202.
• [8] J. Pečarić, T. Furuta, J. M. Hot, and Y. Seo, Mond–Pečarić Method in Operator Inequalities, Monogr. Inequal. 1, Element, Zagreb, 2005.
• [9] H. Zuo, G. Shi, and M. Fujii, Refined Young inequality with Kantorovich constant, J. Math. Inequal. 5 (2011), no. 4, 551–556.