Sharpness of Wilker and Huygens type inequalities

Chen Chao-Ping, Cheung, Cheung Wing-Sum

Description

We present an elementary proof of Wilker's inequality involving trigonometric functions, and establish sharp Wilker and Huygens type inequalities. Mathematics Subject Classification 2010 : 26D05. We present an elementary proof of Wilker's inequality involving trigonometric functions, and establish sharp Wilker and Huygens type inequalities. Mathematics Subject Classification 2010 : 26D05.

Sujets

• Inequality (mathematics)
• Trigonometric functions

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 9
• Langue: English

Extrait

Chen and CheungJournal of Inequalities and Applications2012,2012:72 http://www.journalofinequalitiesandapplications.com/content/2012/1/72

R E S E A R C H

Sharpness of Wilker inequalities 1* 2 ChaoPing Chen and 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

and

Huygens

type

Open Access

Abstract We present an elementary proof of Wilker’s inequality involving trigonometric functions, and establish sharp Wilker and Huygens type inequalities. Mathematics Subject Classification 2010: 26D05. Keywords:inequalities, trigonometric functions

1. Introduction Wilker in [1] proposed two open problems:

(a) Prove that if 0 <x<π/2, then   2 sinxtanx +>2. x x 1

(b) Find the largest constant c such that   2 sinxtanx 3 +>2 +cxtanx x x

(1)

for 0 <x<π/2. In [2], inequality (1) was proved, and the following inequality     4 2 2 sinxtanx8π 3 3 2 +xtanx<+<2 +xtanxfor 0<x<, (2) πx x45 2   4 2 8 where the constants and are best possible, was also established. π 45 Wilker type inequalities (1) and (2) have attracted much interest of many mathemati cians and have motivated a large number of research papers involving different proofs and various generalizations and improvements (cf. [213] and the references cited therein). A brief survey of some old and new inequalities associated with trigonometric functions can be found in [14]. These include (among other results) Wilker’s inequality.

© 2012 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.