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We prove:
(I) For all integers n ≥ 2 and real numbers x ∈ (0,π) we have
,
with the best possible constant bounds
α = (15-√2073)/10240 √(1998-10√2073) = -0.1171..., β = 1/3.
(II) The inequality
holds for all even integers n ≥ 2 and x ∈ (0,π), and also for all odd integers n ≥ 3 and x ∈ (0,π - π/n].
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