This work provides a methodology to study the performance bounds for a H∞ optimal control problem with discrete time LTI plants. In particular, we study a model matching problem with a stable scalar plant that possesses a single finite non-minimum phase zero and an arbitrary relative degree. The solution to this problem can be obtained by several methods that frequently lead to the use of numerical techniques. This usually hides the effect that the plant characteristics have in the optimal achievable value of the objective function. Alternatively, in this paper we use Nehari's Theorem in order to obtain an eigenvalue problem that yields the optimal performance achievable with an one degree-of-freedom control loop. Given the structure of the control setup, we transform the eigenvalue problem into a non-linear equation, whose behaviour can be easily determined. The role that plant characteristics play in the achievable performance bound is made evident by this approach.