Some Properties of Furuta Type Inequalities and Applications

This work is to consider Furuta type inequalities and their applications. Firstly, some Furuta type inequalities underA≥B≥0are obtained via Loewner-Heinz inequality; as an application, a proof of Furuta inequality is given without using the invertibility of operators. Secondly, we show a unified satellite theorem of grand Furuta inequality which is an extension of the results by Fujii et al. At the end, a kind of Riccati type operator equation is discussed via Furuta type inequalities.

Another consequence of tanahashi’s argument on best possibility of the grand Furuta inequality

We show that Tanahashi’s argument on best possibility of the grand Furuta inequality has an additional consequence.

Some Operator Inequalities on Chaotic Order and Monotonicity of Related Operator Function

We will discuss some operator inequalities on chaotic order about several operators, which are generalization of Furuta inequality and show monotonicity of related Furuta type operator function.