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*}
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.
Keywords
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