Operator inequalities among arithmetic mean, geometric mean and harmonic mean (Q2928377)

Operator means are non-commutative extensions of scalar averages. The most famous operator means of two positive operators \(A\) and \(B\) are the weighted arithmetic mean \(A\nabla_\nu B=\nu A+(1-\nu)B\), the weighted geometric mean \(A\sharp_\nu B=A^{\frac{1}{2}}\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right)^\nu A^{\frac{1}{2}}\), and the weighted harmonic mean \(A!_\nu B=\left(\nu A^{-1}+(1-\nu)B^{-1}\right)^{-1}\).NEWLINENEWLINEIn the present paper, the author establishes some inequalities involving these operator means.