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Linear systems subject to nonlinear time-varying sector-bounded uncertainties are considered. Robust performance is measured in terms of the worst-case ratio between the L∞-norms of the output and input signals. A new class of multipliers is introduced and characterized. The characterization can be used to compute an upper bound for robust performance, which tightens the bound obtained by the scaled small gain theorem. While the standard Popov multipliers and their extensions have been developed within the L2 framework, our multipliers are adapted to L∞.