Some Additive Inequalities for Heinz Operator Mean

Abstract

In this paper we obtain some new additive inequalities for Heinz operator mean, namely the operator $H_{\nu }\left( A,B\right) :=\frac{1}{2}\left( A\sharp _{\nu }B+A\sharp _{1-\nu }B\right) $ where $A\sharp _{\nu }B:=A^{1/2}\left( A^{-1/2}BA^{-1/2}\right) ^{\nu }A^{1/2}$ is the weighted geometric mean for the positive invertible operators $A$ and $B,$ and $\nu \in \left[ 0,1\right] .$

Keywords

• Young’s Inequality,Real functions,Arithmetic mean-Geometric mean inequality

References

1. [1] S. S. Dragomir, Some new reverses of Young’s operator inequality, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 130. [http://rgmia.org/papers/v18/v18a130.pdf].
2. [2] S. S. Dragomir, On new refinements and reverses of Young’s operator inequality, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 135. [http://rgmia.org/papers/v18/v18a135.pdf].
3. [3] S. S. Dragomir, Some inequalities for operator weighted geometric mean, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 139. [http://rgmia.org/papers/v18/v18a139.pdf ].
4. [4] S. S. Dragomir, Refinements and reverses of Holder-McCarthy operator inequality, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 143. [http://rgmia.org/papers/v18/v18a143.pdf].
5. [5] S. S. Dragomir, Some reverses and a refinement of H¨older operator inequality, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 147. [http://rgmia.org/papers/v18/v18a147.pdf].
6. [6] S. S. Dragomir, Some inequalities for Heinz operator mean, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 163. [Online http://rgmia.org/papers/v18/v18a163.pdf].
7. [7] S. S. Dragomir, Further inequalities for Heinz operator mean, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 167. [Online http://rgmia.org/papers/v18/v18a167.pdf].
8. [8] S. Furuichi, On refined Young inequalities and reverse inequalities, J. Math. Inequal. 5 (2011), 21-31.

9. [9] S. Furuichi, Refined Young inequalities with Specht’s ratio, J. Egyptian Math. Soc. 20 (2012), 46–49.
10. [10] F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrix, J. Math. Anal. Appl. 361 (2010), 262-269.
11. [11] F. Kittaneh and Y. Manasrah, Reverse Young and Heinz inequalities for matrices, Lin. Multilin. Alg., 59 (2011), 1031-1037.
12. [12] F. Kittaneh, M. Krnic, N. Lovricevic and J. Pecaric, Improved arithmetic-geometric and Heinz means inequalities for Hilbert space operators, Publ. Math. Debrecen, 2012, 80(3-4), 465–478.
13. [13] M. Krnic and J. Pecaric, Improved Heinz inequalities via the Jensen functional, Cent. Eur. J. Math. 11 (9) 2013,1698-1710.
14. [14] F. Kubo and T. Ando, Means of positive operators, Math. Ann. 264 (1980), 205–224.
15. [15] W. Liao, J. Wu and J. Zhao, New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J. Math. 19 (2015), No. 2, pp. 467-479.
16. [16] W. Specht, Zer Theorie der elementaren Mittel, Math. Z. 74 (1960), pp. 91-98.
17. [17] M. Tominaga, Specht’s ratio in the Young inequality, Sci. Math. Japon., 55 (2002), 583-588.H.
18. [18] G. Zuo, G. Shi and M. Fujii, Refined Young inequality with Kantorovich constant, J. Math. Inequal., 5 (2011), 551-556.