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Notes on operator inequalities and positive multilinear mappings | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 09 تیر 1404 اصل مقاله (371.94 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2024.30483.4412 | ||
| نویسنده | ||
| Leila Nasiri* | ||
| Department Mathematics, Lorestan University, Iran | ||
| تاریخ دریافت: 08 اردیبهشت 1402، تاریخ پذیرش: 01 آبان 1403 | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Matrix means؛ Kantorovich's constant؛ Positive multilinear mapping؛ Dual؛ Adjoint | ||
| مراجع | ||
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[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. | ||
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