| In this paper, we study generalization of Furuta inequality and the application of the Furuta inequality 's relative problem. In 1987, Furuta established an operator inequality which is beautiful and historical extension of the famous L(o|¨)wner-Heinz inequality.In the first chapter of this paper, we mainly introduce the history and current state of Furuta inequality and its research backgroud, then we cite some examples to introduce the application of Furuta inequality.The second chapter discusses the monotone operator functions. Firstly, we introduce its definition, then we introduce concrete examples of operator monotone functions obtained by an elementary method without appealing to L(o|¨)wner integral representation. By constructing two groups of operator functions ,we demonstrate the monotonicity of these function under the chaotic order. Futhermore, we study the monotonicity of sequens of operator functions under the chaotic order .The third chapter mainly concentrates on some problems of the Furuta inequality. Firstly, we introduce the geometric structure of Furuta inequality and then discuss the monotonicity of some operator generalized power mean functions under the order defined by A2≥(AB4A)1/3. In addition, we study the Furuta inequality in C*-algebra.In the last chapter, we give a summary of this paper and list some fields worthy of studying about the Furuta inequality and expect more results of Furuta inequality.
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