This work is to consider Furuta type inequalities and their applications. Firstly, some Furuta type inequalities under $A\ge B\ge 0$ are 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.

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