A further refinement of a Jordan type inequality and its application Shan-He Wu a , H.M. Srivastava b, * a Department of Mathematics, Longyan College, Longyan, Fujian 364012, People’s Republic of China b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada Abstract By introducing Taylor polynomials, a new sharpened and generalized version of Jordan’s inequality is established. The result is then used to obtain a substantially more refined inequality of Jordan type. Moreover, an application of the results presented here toward the improvement of the Yang Le inequality is also considered in this paper. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Jordan’s inequality; Taylor polynomials; Higher order derivatives; Yang Le inequality; Generalizations and sharpening of Jordan’s inequality 1. Introduction The following inequality: 2 p 6 sin x x < 1 0 < x 6 p 2   ð1Þ is known in the literature as Jordan’s inequality (see [1], p. 33). Jordan’s inequality (1) and its improved versions have many important applications in calculus, trigonometry and the theory of limits. In recent years, this classical inequality (1) has attracted interest of many mathematicians, and it has been improved as well as generalized in several different directions (see [2–14] and the references cited in them). Recently, Wu and Debnath [15] established the following improved version of Jordan’s inequality (1) by introducing a parameter h: sin h h þ h cos h  sin h h 2 ðx  hÞþ h  2 sin h þ h cos h h 3 ðx  hÞ 2 6 sin x x 6 sin h h þ h cos h  sin h h 2 ðx  hÞþ 2 sin h  2h cos h  h 2 sin h 2h 3 ðx  hÞ 2 ð0 < x 6 h 6 pÞ: ð2Þ 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.08.022 * Corresponding author. E-mail addresses: wushanhe@yahoo.com.cn (S.-H. Wu), harimsri@math.uvic.ca (H.M. Srivastava). Available online at www.sciencedirect.com Applied Mathematics and Computation 197 (2008) 914–923 www.elsevier.com/locate/amc