Document Type : Research Paper


1 Department of Mathematics and computer science, Faculty of science, Lorestan University, Khorramabad, Iran. Khorramabad, Iran.niversity

2 Department of Statistics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran.


In this note, first the better refinements of Young and its reverse inequalities for scalars are given. Then, several operator and norm versions according to these inequalities are established.


Main Subjects

[1] S. Furuichi, On refined Young inequalities and reverse inequalities, J. Math. Inequal., 5 (1) (2011), pp. 21-31.
[2] S. Furuichi, Refined Young inequalities with Specht$^,$s ratio, J. Egyptian Math. Soc., 20 (2012), pp. 46-49.
[3] T. Furuta and M. Yanagida, Generalized means and convexity of inversion for positive operators, Amer. Math. Monthly, 105 (1998), pp. 258-259.
[4] T. Furuta, Invitation to Linear Operators: From Matrix to Bounded Linear Operators on a Hilbert Space, Taylor and Francis, 2002.
[5] T. Furuta and J.M. Hot, Mond-Pecaric Method in operator inequalities, Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Zagreb: Element. 2005.
[6] C.J. He and L.M. Zou, Some inequalities involving unitarily invariant norms, Math. Inequal. Appl., 12 (4) (2012), pp. 767-776.
[7] O. Hirzallah and F. Kittaneh, Matrix Young inequalities for the Hilbert-Schmidt norm, Linear Algebra Appl., 308 (2000), pp. 77-84.
[8] F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl., 361 (2010), pp. 262-269.
[9] F. Kittaneh and Y. Manasrah, Reverse Young and Heinz inequalities for matrices, Linear Multilinear A. 59 (9) ( 2011), pp. 1031-1037.
[10] F. Kittaneh, M. Krnic, N. Lovricevic and J. Pecaric, Improved arithmetic-geometric and Heinz means inequalities for Hilbert space operators, Publ. Math. Debrecen 80 (3-4) (2012), pp. 465-478. 
[11] R. Mikic; J. Pečarić, Inequalities of Ando's Type for $n$-convex Functions , Sahand Commun. Math. Anal., 17 (2020), pp. 139-159.
[12] L. Nasiri and M. Shakoori, Reverses of the Young type inequalities with the classical Kantorovich constant, Results Math. 74 (16) (2019), pp.1-10.
[13] L. Nasiri and W. Liao, The new reverses of Young type inequalities for numbers, operators and matrices, Oper. Matrices, 12 (4) (2018), pp. 1063-1071.
[14] J. Zhao and J. Wu, Operator inequalities involving improved Young and its reverse inequalities, J. Math. Anal. Appl., 421 (2) (2015), pp. 1779-1789.