Sharp Cusa and Becker-Stark inequalities

Chen Chao-Ping, Cheung, Cheung Wing-Sum

Description

We determine the best possible constants θ,ϑ,α and β such that the inequalities 2 + cos x 3 θ < sin x x < 2 + cos x 3 ϑ and π 2 π 2 - 4 x 2 α < tan x x < π 2 π 2 - 4 x 2 β are valid for 0 < × < π/ 2. Our results sharpen inequalities presented by Cusa, Becker and Stark. Mathematics Subject Classification (2000): 26D05. We determine the best possible constants θ,ϑ,α and β such that the inequalities 2 + cos x 3 θ < sin x x < 2 + cos x 3 ϑ and π 2 π 2 - 4 x 2 α < tan x x < π 2 π 2 - 4 x 2 β are valid for 0 < × < π/ 2. Our results sharpen inequalities presented by Cusa, Becker and Stark. Mathematics Subject Classification (2000): 26D05.

Sujets

• Inequality (mathematics)
• Trigonometric functions

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 10
• Langue: English

Extrait

Chen and CheungJournal of Inequalities and Applications2011,2011:136 http://www.journalofinequalitiesandapplications.com/content/2011/1/136

R E S E A R C HOpen Access Sharp Cusa and BeckerStark inequalities 1* 2 ChaoPing Chenand WingSum Cheung

* Correspondence: [e-mail] 1 School of Mathematics and Informatics, Henan Polytechnic, University, Jiaozuo City 454003, Henan Province, People’s Republic of China Full list of author information is available at the end of the article

Abstract We determine the best possible constantsθ,ϑ,aandbsuch that the inequalities     θ ϑ 2 + cosx sinx2 + cosx < < 3x3 and   α   β 2 2 π tanxπ < < 2 22 2 π−4x xπ−4x are valid for 0< × <π/2. Our results sharpen inequalities presented by Cusa, Becker and Stark. Mathematics Subject Classification (2000):26D05. Keywords:Inequalities, trigonometric functions

1. Introduction For 0< × <π/2, it is known in the literature that

sinx2 + cosx <. (1) x3 Inequality (1) was first mentioned by the German philosopher and theologian Nico laus de Cusa (14011464), by a geometrical method. A rigorous proof of inequality (1) was given by Huygens [1], who used (1) to estimate the numberπ. The inequality is now known as Cusa’s inequality [25]. Further interesting historical facts about the inequality (1) can be found in [2]. It is the first aim of present paper to establish sharp Cusa’s inequality. Theorem 1.For0< × <π/2,    θ ϑ 2 + cosxsinx 2 + cosx < < (2) 3x3 with the best possible constants ln(π/2) θ1.11373998= =. . .andϑ= 1. ln(3/2)

© 2011 Chen and Cheung; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.