‎In this talk‎, ‎we present several norm and singular value inequalities for $ \tau $-measurable operators‎. ‎In particular‎, ‎the Young and Heinz mean inequalities are examined and survey their state of equality‎.

‎It is noteworthy that‎, ‎we use the method of Bhatia and Davis to extend inequality that it can well known as a generalization and an extension of refinement of Young's inequality for measurable operators which can be stated as follows‎:

\[ \left\| a^\nu z b^{1-\nu} \right\| _p \leq‎ ‎\left\| a z \right\| _p ^\nu \cdot \left\| z b \right\| _p^{1-\nu} \leq \nu \| az \|_p‎ + ‎(1-\nu) \| zb \|_p‎. \]

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‎Finally‎, ‎we will present a refinement of Heinz mean inequalities for measurable operators.

‎This is joint work with Ms‎. ‎Z‎. ‎Maleki‎.

‎This work was supported by the Department of Mathematical Sciences at Isfahan University of Technology‎, ‎Iran‎.