Linear matrix inequality conditions implying conic bounds are developed for stable linear time-invariant systems with input, output, and/or state delays. Combined with the Conic Sector Theorem, this enables the design of output-feedback controllers ensuring closed-loop input-output stability that is robust to certain maximum delays. These contributions are used in a numerical example to design a nearly-optimal controller with guaranteed input-output-stability for a range of delays and improved performance relative to an H-2-optimal controller designed for the nominally un-delayed system.