Sharpness and generalization of Jordan’s inequality and its application

Let θ ≥ 2 be a given real number, and a , b ∈ R be two parameters, and let Q ( x ; a , b , θ ) = 2 π + a ( π θ − ( 2 x ) θ ) + b ( π θ − ( 2 x ) θ ) 2 . We determine the values a = 2 π − θ − 1 θ , b = ( − π 2 + 4 + 4 θ ) π − 2 θ − 1 4 θ 2 , which provide the best approximation: sin x x ≈ Q ( x ; 2 π − θ − 1 θ , ( − π 2 + 4 + 4 θ ) π − 2 θ − 1 4 θ 2 , θ ) , 0 < x ≤ π 2 . Furthermore, we establish a sharp Jordan’s inequality, and then apply it to improve the Yang Le inequality.