Positive real cones in the space H-infinity appear naturally in many optimization problems of control theory and signal processing. Although such problems can be solved by finite-dimensional approximations (e. g., Ritz projection), all such approximations are conservative, providing one-sided bounds for the optimal value. In order to obtain both upper and lower bounds of the optimal value, a dual problem approach is developed in this paper. A finite-dimensional approximation of the dual problem gives the opposite bound for the optimal value. Thus, by combining the primal and dual problems, a suboptimal solution to the original problem can be found with any required accuracy.