International Journal of Nonlinear Analysis and Applications

Notes on operator inequalities ‎ ‎and‎ positive multilinear mappings

Articles in Press

Document Type : Research Paper

Author

Department Mathematics, Lorestan University, Iran

Abstract

In this paper,  our aim is to prove some matrix inequalities involving arbitrary matrix means and positive multilinear mappings.  For example,   it is  shown  that for  Hermitian  matrices $A_{i}, B_{i}$ such that $0 < m \leq A_{i},B_{i} \leq M \,\,(i=1,\cdots, k),$
\begin{align*}
\Phi^{2} (A_{1} \sigma_{1}  B_{1}, \cdots ,A_{k} \sigma_{1}  B_{k} ) \leq
\left(K(u^{k})\right)^{2}
\Phi^{2} (A_{1} \sigma_{2}  B_{1}, \cdots ,A_{k} \sigma_{2}  B_{k} ),
\end{align*}
where $\sigma_{1}$ and $ \sigma_{2} $  are two arbitrary matrix means between the arithmetic and harmonic means,  $\Phi:\mathscr{M}_n^k(\mathbb{C}) \rightarrow  \mathscr{M}_l(\mathbb{C})$ is a positive unital multilinear mapping, $u=\frac{M}{m} $  and $K(u)=\frac{(1+u)^{2}}{4 u}$.  We also give the obtained results for the adjoint and the dual of an arbitrary matrix mean.

Keywords

[1] T. Ando and X. Zhan, Norm inequalities related to operator monotone functions, Math. Ann. 315 (1999), 771–780.
[2] M. Bakherad, Refinements of a reversed AM-GM operator inequality, Linear Multilinear Algeb. 64 (2016), no. 9, 1687–1695.
[3] R. Bhatia and F. Kittaneh, Notes on matrix arithmetic-geometric mean inequalities, Linear Algebra Appl. 308 (2000), no. 1-3, 203–211.
[4] M. Dehghani, M. Kian, and Y. Seo, Developed matrix inequalities via positive multilinear mappings, Linear Algebra Appl. 484 (2015), 63–85.
[5] X. Fu and C. He, Some operator inequalities for positive linear maps, Linear Multilinear Algeb. 63 (2015), no. 3, 571–577.
[6] X. Fu and D.T. Hoa, On some inequalities with matrix means, Linear Multilinear Algeb. 63(2015), no. 12, 2373–2378.
[7] T. Furuta, J. Micic Hot, J. Pecaric, and Y. Seo, Mond Pecaric Method in Operator Inequalities, Element, Zagreb, 2005.
[8] D.T. Hoa, D.T. H. Binh, and H.M. Toan, On some inequalities with matrix means, RIMS Kokyukoku, Kyotou Univ. 1893 (2014), no. 5, 67–71.
[9] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
[10] M. Kian and M. Dehghani, Extension of the Kantorovich inequality for positive multilinear mappings, Filomat 31 (2017), no. 20, 6473–6481.
[11] F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1980), 205–224.
[12] M. Lin, Squaring a reverse AM-GM inequality, Studia Math. 215 (2013), no. 2, 187–194.
[13] J. Micic, J. Pecaric, and Y. Seo, Complementary inequalities to inequalities of Jensen and Ando based on the Mond-Pecaric method, Linear Algebra Appl. 318 (2000), 87–107.
[14] L. Nasiri and M. Bakherad, Improvements of some operator inequalities involving positive linear maps via the Kantorovich constant, Houston J. Math. 45 (2019), no. 3, 815–830.
[15] L. Nasiri and W. Liao, The new reverses of Young type inequalities for numbers, operators and matrices, Oper. Matrices 12 (2018), no. 4, 1063-1071.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 30 June 2025
  • Receive Date: 28 April 2023
  • Accept Date: 22 October 2024