Two discrete time H∝mixed sensitivity problems are investigated, and a closed formula derived for the optimal cost associated with each. Our approach is based on a state-space analysis, and involves solving two H∝Riccati equations in terms of the stabilizing solutions to associated H2Riccati equations. This not only enables the optimal cost to be directly obtained, but also allows considerable simplification of the process of generating controllers which satisfy any desired suboptimal bound. The main result is proved using conditions given with a recent 2-Riccati-equation characterization of discrete time controllers satisfying a closed loop H∝norm bound.