Improved bounds for and the factorical polynomial.
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].