Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Sharp Becker-Stark-Type Inequalities for Bessel Functions | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 838740 4 pages doi 2010 838740 Research Article Sharp Becker-Stark-Type Inequalities for Bessel Functions Ling Zhu Department of Mathematics Zhejiang Gongshang University Hangzhou Zhejiang 310018 China Correspondence should be addressed to Ling Zhu zhuling0571@ Received 22 January 2010 Accepted 23 March 2010 Academic Editor Wing-Sum Cheung Copyright 2010 Ling Zhu. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We extend the Becker-Stark-type inequalities to the ratio of two normalized Bessel functions of the first kind by using Kishore formula and Rayleigh inequality. 1. Introduction In 1978 Becker and Stark 1 or see Kuang 2 page 248 obtained the following two-sided rational approximation for tan x x. Theorem . Let 0 x n 2 then 8 tan x n2 n2 - 4x2 x n2 - 4x2 Furthermore 8 and n2 are the best constants in . In recent paper 3 we obtained the following further result. Theorem . Let 0 x n 2 then n2 4 8 - n2 n2 x2 tanx n2 n2 3 - 4 x2 n2 - 4x2 x n2 - 4x2 Furthermore a 4 8 - n2 n2 and p n2 3 - 4 are the best constants in . 2 Journal of Inequalities and Applications Moreover the following refinement of the Becker-Stark inequality was established in 3 . Theorem . Let 0 x n 2 and N 0 be a natural number. Then P2N x ax2N 2 tan x P2N x px2N 2 1 3 n2 - 4x2 x n2 - 4x2 holds where P2N x a0 a1x2 aNx2N and 22n 2 22n 2 - 1 n2 4 22n 22n -1 an ---- 2n 2 I B2n 21---- ny B2n n 0 1 2. L4 where B2n are the even-indexed Bernoulli numbers. Furthermore p aN 1 and a 8 - a0 -a1 n 2 2-------aN n 2 2N n 2 2N 2 are the best constants in . Our aim of this paper is to extend the tangent function to Bessel functions. To achieve our goal let us recall some basic facts about Bessel functions. Suppose .
