Document Type : Research Paper

Authors

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

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

Abstract

In this paper, we establish further improvements  of the Young inequality and its reverse. Then, we assert operator versions corresponding them. Moreover, an application including positive linear mappings is given. For example, if $A,B\in {\mathbb B}({\mathscr H})$ are two invertible positive operators such that 0\begin{align*} & \Phi ^{2} \bigg(A \nabla _{\nu} B+ rMm \left( A^{-1}+A^{-1} \sharp_{\mu} B^{-1} -2 \left(A^{-1} \sharp_{\frac{\mu}{2}} B^{-1} \right)\right)\\ & \qquad +\left(\frac{\nu}{\mu} \right) Mm \bigg(A^{-1}\nabla_{\mu} B^{-1} -A^{-1} \sharp_{\mu} B^{-1} \bigg)\bigg) \\ & \quad \leq \left( \frac{K(h)}{ K\left( \sqrt{{h^{'}}^{\mu}},2 \right)^{r^{'}}} \right) ^{2} \Phi^{2} (A \sharp_{\nu} B), \end{align*} wherer=\min\{\nu,1-\nu\}$,$K(h)=\frac{(1+h)^{2}}{4h}$,$h=\frac{M}{m}$,$h^{'}=\frac{M^{'}}{m^{'}}$and$r^{'}=\min\{2r,1-2r\}$. The results of this paper generalize the results of recent years. Keywords 20.1001.1.23225807.2022.19.2.9.6 ###### ##### References  M. Bakherad, Refinements of a reversed AM-GM operator inequality, Linear Multilinear Algebra, 64 (2016), pp. 1687-1695.  R. Bhatia, Positive Definite Matrices, Princeton University Press, Princeton, 2007.  R. Bhatia and F. Kittaneh, Notes on matrix arithmetic-geometric mean inequalities, Linear Algebra Appl., 308 (2000), pp. 203-211.  T. Furuta, J. Micic Hot, J. Pecaric and Y. Seo, Mond-Pecaric method in operator inequalities, Element, Zagreb, 2005.  F. Kubo and T. Ando, Means of positive linear operators, Math. Ann., 246 (1980), pp. 205-224.  M. Lin, Squaring a reverse AM-GM inequality, Stud. Math., 215 (2013), pp. 187-194.  W. Liao, J. Wu and S. Zhao, New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J. Math., 19 (2015), pp. 467-479.  R. Mikic; J. Pečarić, Inequalities of Ando's Type for$n\$-convex Functions , Sahand Commun. Math. Anal., 17 (2020), pp. 139-159.
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