arXiv subjects
Publications and source records attributed to I. H. Gümüş.
We show the following result: Let $A$ be a positive operator satisfying $0<m{\mathbf{1}_{\mathcal{H}}}\le A\le M{\mathbf{1}_{\mathcal{H}}}$ for some scalars $m,M$ with $m<M$ and $Φ$ be a normalized positive linear map, then \[Φ\left( {{A}^{-1}} \right)\le Φ\left( {{m}^{\frac{A-M{\mathbf{1}_{\mathcal{H}}}}{M-m}}}{{M}^{\frac{m{\mathbf{1}_{\mathcal{H}}}-A}{M-m}}} \right)\le \frac{{{\left( M+m \right)}^{2}}}{4Mm}Φ{{\left( A \right)}^{-1}}.\]
We improve and generalize some operator inequalities for positive linear maps. It is shown, among other inequalities, that if $0<m\le B\le m'<M'\le A\le M$ or $0<m\le A\le m'<M'\le B\le M$, then for each $2\le p<\infty $ and $ν\in \left[ 0,1 \right]$, \begin{equation*} {{Φ}^{p}}\left( A{{\nabla }_{ν}}B \right)\le {{\left( \frac{K\left( h \right)}{{{4}^{\frac{2}{p}-1}}{{K}^{r}}\left( h' \right)} \right)}^{p}}{{Φ}^{p}}\left( A{{\#}_{ν}}B \right), \end{equation*} and \begin{equation*} {{Φ}^{p}}\left( A{{\nabla }_{ν}}B \right)\le {{\left( \frac{K\left( h \right)}{{{4}^{\frac{2}{p}-1}}{{K}^{r}}\left( h' \right)} \right)}^{p}}{{\left( Φ\left( A \right){{\#}_{ν}}Φ\left( B \right) \right)}^{p}}, \end{equation*} where $r=\min \left\{ ν,1-ν\right\}$, $h=\frac{M}{m}$ and $h'=\frac{M'}{m'}$. We also obtain an improvement of operator Pólya-Szegö inequality.