Some new inequalities involving Heinz operator means

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.