For a class of discrete-time linear time-invariant (LTI) plants, we present necessary and sufficient conditions for the existence of a discrete-time LTI controller that ensures that the H-infinity norm of a closed-loop map of interest is less than a prespecified level, subject to time-domain constraints on this closed-loop map, In cases when such an LTI controller does not exist, we show that the problem of finding an LTI controller that satisfies the input-output constraints with the smallest error can be posed as a convex optimization problem based on linear matrix inequalities, We also consider, briefly, other related problems that can be solved completely using a similar approach, namely controller synthesis to minimize the optimally seated H-infinity norm and controller synthesis to maximize the guaranteed dissipation.