This paper investigates the design of reduced-order controllers using an H-infinity framework. Given a stabilizing controller which satisfies a prespecified level of closed-loop H-infinity performance, sufficient conditions are derived for another controller to be stabilizing and satisfy the same level of H-infinity performance. Such controllers are said to be (P, gamma)-admissible, where P is the model of the plant under consideration and gamma is the required level of prespecified H-infinity, performance. The conditions are expressed as norm bounds on particular frequency-weighted errors, where the weights are Selected to make a specific transfer function a contraction. The design of reduced-order (P,gamma)-admissible controllers is then formulated as a frequency-weighted model reduction problem.It is advantageous for the required weights to be large in some sense, Solutions which minimize either the trace, or the determinant, of the inverse weights are characterized. We show that the procedure for minimizing the determinant of the inverse weights always gives a direction where the weights are the best possible.To conclude, we demonstrate by way of a numerical example, that when used in conjunction with a combined model reduction/convex optimization scheme, the proposed design procedures are effective in substantially reducing controller complexity.