We give some new refinements of Heinz inequality and an improvement of the reverse Young's inequality for scalars and we use them to establish new inequalities for operators and the Hilbert -Schmidt norm of matrices. We give a uniformly and abbreviated form of the inequalities presented by Kittaneh and Mansarah, and the inequalities presented by Kai and we obtain some of their operator and matrix versions.

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Let A, B be invertible positive operators on a Hilbert space H. We present some improved reverses of Young type inequalities, in particular,andwhere 0 ≤ υ ≤ 1 2 . We also give some new inequalities involving the Heinz mean for the Hilbert-Schmidt norm.