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A classical theorem due to Young states that the cosine polynomial is positive for all n 1 and x ∈ (0, π). We prove the following refinement. For all n 2 and x ∈ [0, π] we have with the best possible constant factor c = min 0 t<π 5 + 6 cos(t) + 3 cos(2t) 6(π − t) 2 = 0.069 . . . .doi:10.1017/s0013091504000744 fatcat:ep2mmcezjrhfve3lxgqycxg7ve