On Some Operator Inequalities

In this dissertation, we mainly consider the refinements or generalizations of some operator inequalities on a Hilbert space. We intend to seperate this dissertation into four chapters.In the first chapter, we introduce some related inequalities and background about operator inequalities. Besides, some theorems and knowledge are also known.In the second chapter, we study operator inequalities about operator monotone func-tion. The results of T.Furuta and M.Moslehian are improved and extended.In the third chapter, we estimate bounds of operator function convexity(concavity) by Jensen inequality. Based on this we obtain some relations between the power of arithmetic(geometric) mean and the arithmetic(geometric) mean, by which some theorems of M.Fujii are extended.In the fourth chapter, we introduce some Kantorovich-type inequality on C*-algebra and extend some covariance-variance inequalities.