A new refinement of Young's inequality
SourceProceedings of the Edinburgh Mathematical Society
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A classical theorem due to Young states that the cosine polynomial c n(x) = 1 + ∑k=1n cos(kx)/k 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 1/6 + c(π - x)2 ≤ Cn(x), with the best possible constant factor c = min 0≤t<π 5 +6 cos(t) + 3 cos(2t)/6(π - t)2 = 0.069....