A generalization of Jenses's operator inequality from a function of one variable to a function of many variables is obtained and shown to be equivalent to the operator convexity of the function plus a boundary condition that the function is non-positive at the boundary of the domain of the function, which is taken to be the product of intervals [0,a_j) (0 less a_j less or equal infinity). A unitary row, which is defined to be a set of operators forming a row of unitary block matrix, is used for the statement of the generalized Jensen's operator inequality.

The same type of inequality, where unitary rows are replaced by contraction rows, is also obtained as the consequence of the operator convexity plus the boundary condition for functions of several variables.

Joint work with Frank Hansen.