This paper considers the problem of finding the H-infinity-norm of a linear time-varying system having a delayed state and with an output that comprises a linear combination of the "current" state as well as of the "delayed" state. Both delay-independent and delay-dependent bounded real criteria are derived which provide sufficient conditions for the system to possess an H-infinity-norm which is less than or equal to a prescribed bound. The stationary case for time-invariant systems is also tackled, and in this situation the proposed bounded real criteria are given in terms of linear matrix inequalities. Two examples are presented which compare the guaranteed bounds that are achieved by our results with the actual H-infinity-norm of the system.