A new relative error model reduction method is proposed using frequency-weighted balanced realization, and explicit L(infinity) norm error bounds are also derived for the relative error and multiplicative error. The method only needs to solve two Lyapunov equations. It is further shown that this method is equivalent to the balanced stochastic truncation if the plant is square and minimum phase. This paper also gives a complete solution to the frequency-weighted Wankel norm approximation with antistable weighting. These results are then applied to L(infinity) norm model reduction, and several numerically effective algorithms are proposed. It is shown through many numerical examples that these algorithms work very well and in many cases produce almost optimal solutions.