This note proposes a class of the constantly scaled H-infinity control problem, where the class is characterized by an assumption also proposed in this note. In general, the scaled H-infinity control problem leads to nonconvex solvability conditions. On the other hand, for the problems in the proposed class, we show that a convex but sufficient solvability condition can be given. Moreover, the convexity of the state feedback and the full information problems can be regarded as the extreme cases of the sufficient condition. This fact motivates us to call the assumption the pseudofull information condition.