Sharpness of Wilker and Huygens type inequalities
© Chen and Cheung; licensee Springer. 2012
Received: 29 June 2011
Accepted: 28 March 2012
Published: 28 March 2012
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.
Keywordsinequalities trigonometric functions
- (a)Prove that if 0 < x < π/2, then(1)
- (b)Find the largest constant c such that
for 0 < x < π/2.
where the constants and are best possible, was also established.
Wilker type inequalities (1) and (2) have attracted much interest of many mathematicians 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 . These include (among other results) Wilker's inequality.
in terms of the arithmetic, geometric and harmonic means.
In this article, we present an elementary proof of Wilker's inequality (2), and we establish sharp Wilker and Huygens type inequalities.
The following lemma is also needed in the sequel.
Lemma 1.  Let a n ∈ ℝ and b n > 0, n = 0,1, 2,... be real numbers with being strictly increasing (respectively, decreasing). If the power series and are convergent for |x| < R, then the function A(x)/B(x) is strictly increasing (respectively, decreasing) on (0, R).
2. An elementary proof of Wilker's inequality (2)
for all , with the constants and being best possible. This completes the proof of (2).
3. Sharp Wilker's inequality
Motivated by (12), we are now in a position to establish our first main result.
- (ii)For 0 < x < π/2, we have(14)
The constants and are best possible.
Proof. We only prove inequality (13). The proof of (14) is analogous.
for all , with the constants and being best possible. This completes the proof of (13).
Remark 1. Inequality (14) is sharper than (13). On the other hand, there is no strict comparison between inequalities (2) and (13). There is no strict comparison between inequalities (2) and (14) either.
In view of inequalities (13) and (14), we propose the following conjecture.
4. Sharp the Wu-Srivastava inequality
Motivated by (16), we establish our second main result:
- (ii)For 0 < x < π/2, we have(18)
The constant is best possible.
Proof. We only prove inequality (18). The proof of (17) is analogous.
The proof of the inequality (21) is not difficult, and is left with the readers. This proves the claim.
with the constant being best possible. This completes the proof of (18).
In view of inequalities (17) and (18), we propose the following conjecture.
5. Skarp Huygens inequality
Motivated by (23), we establish our third main result:
- (ii)For 0 < x < π/2, we have(25)
The constants and are best possible.
Proof. We only prove inequality (25). The proof of (24) is analogous.
for all with the constants and being possible. This completes the proof of (25).
Remark 2. There is no strict comparison between inequalities (24) and (25).
In view of inequalities (24) and (25), we propose the following conjecture.
The research is supported in part by the Research Grants Council of the Hong Kong SAR, Project No. HKU7016/07P
- Wilker JB: Problem E 3306. Am Math Mon 1989, 96: 55. 10.2307/2323260MathSciNetView ArticleGoogle Scholar
- Sumner JS, Jagers AA, Vowe M, Anglesio J: Inequalities involving trigonometric functions. Am Math Mon 1991, 98: 264–267. 10.2307/2325035MathSciNetView ArticleGoogle Scholar
- Guo BN, Qiao BM, Qi F, Li W: On new proofs of Wilker inequalities involving trigonometric functions. Math Inequal Appl 2003, 6: 19–22.MathSciNetMATHGoogle Scholar
- Mortitc C: The natural approach of Wilker-Cusa-Huygens inequalities. Math Inequal Appl 2011, 14: 535–541.MathSciNetMATHGoogle Scholar
- Neuman E: On Wilker and Huygens type inequalities. Math Inequal Appl, in press.Google Scholar
- Pinelis I: L'Hospital rules of monotonicity and Wilker-Anglesio inequality. Am Math Mon 2004, 111: 905–909. 10.2307/4145099MathSciNetView ArticleMATHGoogle Scholar
- Wu SH, Srivastava HM: A weighted and exponential generalization of Wilker's inequality and its applications. Integr Trans Spec Funct 2007, 18: 529–535. 10.1080/10652460701284164MathSciNetView ArticleMATHGoogle Scholar
- Wu SH, Srivastava HM: A further refinement of Wilker's inequality. Integr Trans Spec Funct 2008, 19: 757–765. 10.1080/10652460802340931MathSciNetView ArticleMATHGoogle Scholar
- Zhang L, Zhu L: A new elementary proof of Wilker's inequalities. Math Inequal Appl 2008, 11: 149–151.MathSciNetMATHGoogle Scholar
- Zhu L: A new simple proof of Wilker's inequality. Math Inequal Appl 2005, 8: 749–750.MathSciNetMATHGoogle Scholar
- Zhu L: On Wilker-type inequalities. Math Inequal Appl 2007, 10: 727–731.MathSciNetMATHGoogle Scholar
- Zhu L: Some new Wilker-type inequalities for circular and hyperbolic functions. Abstr Appl Anal 2009, 2009: 9. (Article ID 485842)MathSciNetMATHGoogle Scholar
- Zhu L: A source of inequalities for circular functions. Comput Math Appl 2009, 58: 1998–2004. 10.1016/j.camwa.2009.07.076MathSciNetView ArticleMATHGoogle Scholar
- Srivastava R: Some families of integral, trigonometric and other related inequalities. Appl Math Inf Sci 2011, 5: 342–360.MathSciNetGoogle Scholar
- Huygens C: Oeuvres Completes 1888–1940. Société Hollondaise des Science, Haga;Google Scholar
- Neuman E, Sándor J: On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities. Math Inequal Appl 2010, 13: 715–723.MathSciNetMATHGoogle Scholar
- Zhu L: Some new inequalities of the Huygens type. Comput Math Appl 2009, 58: 1180–1182. 10.1016/j.camwa.2009.07.045MathSciNetView ArticleMATHGoogle Scholar
- Baricz A, Sándor J: Extensions of generalized Wilker inequality to Bessel functions. J Math Inequal 2008, 2: 397–406.MathSciNetView ArticleMATHGoogle Scholar
- Mitrinović DS: Analytic Inequalities. Springer-Verlag, Berlin; 1970.View ArticleGoogle Scholar
- Chen CP, Cheung WS: Wilker- and Huygens-type inequalities and solution to Oppenheim's problem. Integr Trans Spec Funct, in press.Google Scholar
- Ponnusamy S, Vuorinen M: Asymptotic expansions and inequalities for hypergeometric functions. Mathematika 1997, 44: 278–301. 10.1112/S0025579300012602MathSciNetView ArticleMATHGoogle Scholar
- Abramowitz M, Stegun IA: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. In Appl Math Ser National Bureau of Standards. Volume 55. 9th edition. Washington, D.C; 1972.Google Scholar
- Kuang J-Ch: Applied Inequalities. 3rd edition. Shandong Science and Technology Press, Jinan City, Shandong Province, China (Chinese); 2004.Google Scholar
This article is published under license to BioMed Central Ltd. 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.