Abstract
The main purpose of this paper is to present some weighted arithmetic-geometric operator mean inequalities. These inequalities are refinements and generalizations of the corresponding results. An example is provided to confirm the effectiveness of the results.
Highlights
It is well known that A ≥ B implies A2 ≥ B2 for some positive operators A, B
We show new weighted arithmetic-geometric operator mean inequalities which generalize inequalities (3) and (4)
We first present two weighted arithmetic-geometric operator mean inequalities, which refine and generalize inequalities (5) and (6), an example shows that inequalities (11)
Summary
It is well known that A ≥ B implies A2 ≥ B2 for some positive operators A, B. It is interesting to ask for what kind of operator inequalities, when they are squared, the inequality relation can be preserved. In 2013, Lin ([1], Theorem 2.8) proved that the operator Kantorovich inequality can be squared. Lin ([2], Theorem 2.1) found that the reverse arithmetic-geometric operator mean inequalities for the Kantorovich constant can be squared: Φ2 and. Is called the Kantorovich constant and satisfies the following properties:
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