arXiv subjects
Publications and source records attributed to Z. Heydarbeygi.
In this article, we explore the celebrated Grüss inequality, where we present a new approach using the Grüss inequality to obtain new refinements of operator means inequalities. We also present several operator Grüss-type inequalities with applications to the numerical radius and entropies.
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 \[Φ\left( {{A}^{-1}} \right)\le Φ\left( {{m}^{\frac{A-M{\mathbf{1}_{\mathcal{H}}}}{M-m}}}{{M}^{\frac{m{\mathbf{1}_{\mathcal{H}}}-A}{M-m}}} \right)\le \frac{{{\left( M+m \right)}^{2}}}{4Mm}Φ{{\left( A \right)}^{-1}}.\]