The Generalized Inequalities via Means and Positive Linear Mappings
Journal: Sahand Communications in Mathematical Analysis (Vol.19, No. 2)Publication Date: 2022-06-15
Authors : Leila Nasiri; Mehdi Shams;
Page : 133-148
Keywords : Operator means; Numerical means; Kantorovich's constant; Positive linear map;
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 $A,Bin {mathbb B}({mathscr H})$ are two invertible positive operators such that $0begin{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)}{ Kleft( sqrt{{h^{'}}^{mu}},2 right)^{r^{'}}} right) ^{2} Phi^{2} (A sharp_{nu} B),
end{align*}
where $r=min{nu,1-nu}$, $K(h)=frac{(1+h)^{2}}{4h}$, $h=frac{M}{m}$, $h^{'}=frac{M^{'}}{m^{'}}$ and $r^{'}=min{2r,1-2r}$. The results of this paper generalize the results of recent years.
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Last modified: 2022-07-31 17:27:17