Some inequalities for interpolational operator means
arXiv:1808.08342
Abstract
Using the properties of geometric mean, we shall show for any , \[f\left( A{{\nabla }_{α}}B \right)\le f\left( \left( A{{\nabla }_{α}}B \right){{\nabla }_{β}}A \right){{\sharp}_{α}}f\left( \left( A{{\nabla }_{α}}B \right){{\nabla }_{β}}B \right)\le f\left( A \right){{\sharp}_{α}}f\left( B \right)\] whenever is a non-negative operator log-convex function, are positive operators, and . As an application of this operator mean inequality, we present several refinements of the Aujla subadditive inequality for operator monotone decreasing functions. Also, in a similar way, we consider some inequalities of Ando's type. Among other things, it is shown that if is a positive linear map, then \[Φ\left( A{{\sharp}_{α}}B \right)\le Φ\left( \left( A{{\sharp}_{α}}B \right){{\sharp}_{β}}A \right){{\sharp}_{α}}Φ\left( \left( A{{\sharp}_{α}}B \right){{\sharp}_{β}}B \right)\le Φ\left( A \right){{\sharp}_{α}}Φ\left( B \right).\]
9 pages