Sharpening Jordan’s inequality and the Yang Le inequality

In this work, the following inequality: sinxx≤2π+π−2π3(π2−4x2),x∈(0,π/2] is established. An application of this inequality gives an improvement of the Yang Le inequality [C.J. Zhao, Generalization and strengthening of the Yang Le inequality, Math. Practice Theory 30 (4) (2000) 493–497 (in Chinese)]:(n−1)∑k=1ncos2λAk−2cosλπ∑1≤i0(i=1,2,…,n),∑i=1nAi≤π,0≤λ≤1, and n≥2n≥2 is a natural number.