Cauchy inequality and uncertainty relations for mixed states

Cauchy inequality (CI) relates scalar products of two vectors and their norms. I point out other similar inequalities (SI). Starting with CI Schroedinger (1930) derived his uncertainty relation (UR). By using SI other various UR can be obtained. It is shown that they follow from the Schroedinger UR. Two generalizations of CI are obtained for mixed states described by density matrices. Using them two generalizations of UR for mixed states are derived. Both differ from the UR generalization known in the literature. The discussion of these generalizations is given.