A generalization of Jordan's inequality and an application

Abstract

In this article, a new generalization of Jordan’s inequality $sumlimits_{k=1}^n mu_k (theta^t -x^t)^k leq frac{sin x} {x} - frac{sin theta} {theta} leq sumlimits_{k=1}^n omega_k (theta^t -x^t)^k$ for $t geq 2, n in Bbb{N}$ and $theta in (0, pi ]$ is established, where the coefficients $mu_k$ and $omega_k$ are defined by recursion formulas, and are the best possible. As an application, Yang’s inequality is refined.