/ / Article

Research Article | Open Access

Volume 2009 |Article ID 485842 | 9 pages | https://doi.org/10.1155/2009/485842

# Some New Wilker-Type Inequalities for Circular and Hyperbolic Functions

Accepted11 May 2009
Published09 Jul 2009

#### Abstract

In this paper, we give some new Wilker-type inequalities for circular and hyperbolic functions in exponential form by using generalizations of Cusa-Huygens inequality and Cusa-Huygens-type inequality.

#### 1. Introduction

Wilker  proposed two open questions, the first of which was the following statement.

Problem 1. Let . Then holds.

Sumner et al.  proved inequality (1.1). Guo et al.  gave a new proof of inequality (1.1). Zhu [4, 5] showed two new simple proofs of Wilker’s inequality above, respectively.

Recently, Wu and Srivastava  obtained Wilker-type inequality as follows: Baricz and Sandor  found that inequality (1.2) can be proved by using inequality (1.1).

On the other hand, in the form of inequality (1.1), Zhu  obtained the following Wilker type inequality:

In fact, we can obtain further results:

In this note, we establish the following four new Wilker type inequalities in exponential form for circular and hyperbolic functions.

Theorem 1.1. Let , and . Then(i)when , the inequality holds;(ii)when , inequality (1.5) is revered.

Theorem 1.2. Let and . Then the inequality holds.

Theorem 1.3. Let , and . Then(i)when , the inequality holds;(ii)when , inequality (1.7) is revered.

Theorem 1.4. Let and . Then the inequality holds.

#### 2. Lemmas

Lemma 2.1 (see ). Let be two continuous functions which are differentiable on . Further, let on . If is increasing (or decreasing) on , then the functions and are also increasing (or decreasing) on .

Lemma 2.2 (see ). Let and () be real numbers, and let the power series and be convergent for . If for and if is strictly increasing (or decreasing) for then the function is strictly increasing (or decreasing) on .

Lemma 2.3 (see [28, 29]). Let , then the inequality holds.

Lemma 2.4. Let , then the inequality holds.

Proof. The following power series expansion can be found in [30, 1.3.1.4 ( 3)] Then

Lemma 2.5 (see [5, 31]). Let . Then the inequality holds.

Lemma 2.6 (see [5, 31, 32]). Let . Then the inequality holds.

Lemma 2.7. Let . Then the function increases as increases on .

Lemma 2.8. Let . Then the function increases as increases on .

Lemma 2.9 (a generalization of Cusa-Huygens inequality). Let and . Then the inequality or holds.

Lemma 2.10 (a generalization of Cusa-Huygens type inequality). Let and . Then the inequality or holds.

#### 3. Proofs of Lemma 2.7 and Theorem 1.1

Proof of Lemma 2.7. Direct calculation yields , where First, we have by Lemma 2.5. Second, when letting for , we have , and for , so and . Thus and . The proof of Lemma 2.7 is complete.

Proof of Theorem 1.1. From Lemma 2.7 we have for . That is, (1.5) holds. At the same time, we have for . That is, (1.5) is revered.

#### 4. Proofs of Lemma 2.9 and Theorem 1.2

Proof of Lemma 2.9. Let , where , and . Then where where . Then where , and . By (2.1), (2.2), and (2.3), we have where and .
When setting , we have that is increasing for is increasing from onto by Lemma 2.2. When , we have . So is increasing on . This leads to that is increasing on . Thus is increasing on by Lemma 2.1. At the same time, . So the proof of Lemma 2.9 is complete.

Proof of Theorem 1.2. From Theorem 1.1, when we have On the other hand, when we can obtain by the arithmetic mean-geometric mean inequality and Lemma 2.9. So holds.
Combining (4.5) and (4.7) gives (1.6).

#### 5. Proofs of Lemma 2.8 and Theorem 1.3

Proof of Lemma 2.8. Direct calculation yields , where First, for . Second, we have by Lemma 2.6. Thus and . The proof of Lemma 2.8 is complete.

Proof of Theorem 1.3. From Lemma 2.8 we have for . That is, (1.7) holds. At the same time, we have for . That is, (1.7) is revered.

#### 6. Proofs of Lemma 2.10 and Theorem 1.4

Proof of Lemma 2.10. Let , where , and . Then where and . Since where and .
When setting , we have is decreasing for is decreasing on by Lemma 2.2. At the same time, the function is decreasing on when . By (6.1), we obtain that is decreasing on . Thus is decreasing on by Lemma 2.1. At the same time, . So the proof of Lemma 2.10 is complete.

Proof of Theorem 1.4. By the same way as Theorem 1.2, we can prove Theorem 1.4.

#### 7. Open Problem

In this section, we pose the following open problem: find the respective largest range of such that the inequalities (1.6) and (1.8) hold.

1. J. B. Wilker, “E3306,” The American Mathematical Monthly, vol. 96, no. 1, p. 55, 1989. View at: Google Scholar
2. J. S. Sumner, A. A. Jagers, M. Vowe, and J. Anglesio, “Inequalities involving trigonometric functions,” The American Mathematical Monthly, vol. 98, no. 3, pp. 264–267, 1991. View at: Google Scholar
3. B.-N. Guo, B.-M. Qiao, F. Qi, and W. Li, “On new proofs of Wilker's inequalities involving trigonometric functions,” Mathematical Inequalities & Applications, vol. 6, no. 1, pp. 19–22, 2003.
4. L. Zhu, “A new simple proof of Wilker's inequality,” Mathematical Inequalities & Applications, vol. 8, no. 4, pp. 749–750, 2005.
5. L. Zhu, “On Wilker-type inequalities,” Mathematical Inequalities & Applications, vol. 10, no. 4, pp. 727–731, 2007.
6. S.-H. Wu and H. M. Srivastava, “A weighted and exponential generalization of Wilker's inequality and its applications,” Integral Transforms and Special Functions, vol. 18, no. 7-8, pp. 529–535, 2007.
7. A. Baricz and J. Sandor, “Extensions of the generalized Wilker inequality to Bessel functions,” Journal of Mathematical Inequalities, vol. 2, no. 3, pp. 397–406, 2008. View at: Google Scholar
8. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, “Inequalities for quasiconformal mappings in space,” Pacific Journal of Mathematics, vol. 160, no. 1, pp. 1–18, 1993.
9. G. D. Anderson, S.-L. Qiu, M. K. Vamanamurthy, and M. Vuorinen, “Generalized elliptic integrals and modular equations,” Pacific Journal of Mathematics, vol. 192, no. 1, pp. 1–37, 2000.
10. I. Pinelis, “L'Hospital type results for monotonicity, with applications,” Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 1, article 5, pp. 1–5, 2002.
11. I. Pinelis, ““Non-strict” l'Hospital-type rules for monotonicity: intervals of constancy,” Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 1, article 14, pp. 1–8, 2007. View at: Google Scholar | MathSciNet
12. L. Zhu, “Sharpening Jordan's inequality and the Yang Le inequality,” Applied Mathematics Letters, vol. 19, no. 3, pp. 240–243, 2006.
13. L. Zhu, “Sharpening Jordan's inequality and Yang Le inequality. II,” Applied Mathematics Letters, vol. 19, no. 9, pp. 990–994, 2006.
14. L. Zhu, “Sharpening of Jordan's inequalities and its applications,” Mathematical Inequalities & Applications, vol. 9, no. 1, pp. 103–106, 2006.
15. L. Zhu, “Some improvements and generalizations of Jordan's inequality and Yang Le inequality,” in Inequalities and Applications, Th. M. Rassias and D. Andrica, Eds., CLUJ University Press, Cluj-Napoca, Romania, 2008. View at: Google Scholar
16. L. Zhu, “A general refinement of Jordan-type inequality,” Computers & Mathematics with Applications, vol. 55, no. 11, pp. 2498–2505, 2008.
17. F. Qi, D.-W. Niu, J. Cao, and S. X. Chen, “A general generalization of Jordan's inequality and a refinement of L. Yang's inequality,” RGMIA Research Report Collection, vol. 10, no. 3, supplement, 2007. View at: Google Scholar
18. S. Wu and L. Debnath, “A new generalized and sharp version of Jordan's inequality and its applications to the improvement of the Yang Le inequality,” Applied Mathematics Letters, vol. 19, no. 12, pp. 1378–1384, 2006.
19. S. Wu and L. Debnath, “A new generalized and sharp version of Jordan's inequality and its applications to the improvement of the Yang Le inequality. II,” Applied Mathematics Letters, vol. 20, no. 5, pp. 532–538, 2007. View at: Google Scholar | MathSciNet
20. D.-W. Niu, Z.-H. Huo, J. Cao, and F. Qi, “A general refinement of Jordan's inequality and a refinement of L. Yang's inequality,” Integral Transforms and Special Functions, vol. 19, no. 3-4, pp. 157–164, 2008.
21. S.-H. Wu, H. M. Srivastava, and L. Debnath, “Some refined families of Jordan-type inequalities and their applications,” Integral Transforms and Special Functions, vol. 19, no. 3-4, pp. 183–193, 2008.
22. S. Wu and L. Debnath, “A generalization of L'Hospital-type rules for monotonicity and its application,” Applied Mathematics Letters, vol. 22, no. 2, pp. 284–290, 2009. View at: Google Scholar | MathSciNet
23. S. Wu and L. Debnath, “Jordan-type inequalities for differentiable functions and their applications,” Applied Mathematics Letters, vol. 21, no. 8, pp. 803–809, 2008. View at: Publisher Site | Google Scholar
24. S. Wu and L. Debnath, “A generalization of L'Hôspital-type rules for monotonicity and its application,” Applied Mathematics Letters, vol. 22, no. 2, pp. 284–290, 2009. View at: Google Scholar | MathSciNet
25. M. Biernacki and J. Krzyż, “On the monotonity of certain functionals in the theory of analytic functions,” Annales Universitatis Mariae Curie-Sklodowska, vol. 9, pp. 135–147, 1955. View at: Google Scholar | MathSciNet
26. S. Ponnusamy and M. Vuorinen, “Asymptotic expansions and inequalities for hypergeometric functions,” Mathematika, vol. 44, no. 2, pp. 278–301, 1997.
27. H. Alzer and S.-L. Qiu, “Monotonicity theorems and inequalities for the complete elliptic integrals,” Journal of Computational and Applied Mathematics, vol. 172, no. 2, pp. 289–312, 2004.
28. J.-L. Li, “An identity related to Jordan's inequality,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 76782, 6 pages, 2006.
29. S.-H. Wu and H. M. Srivastava, “A further refinement of Wilker's inequality,” Integral Transforms and Special Functions, vol. 19, no. 9-10, pp. 757–765, 2008. View at: Publisher Site | Google Scholar | MathSciNet
30. A. Jeffrey, Handbook of Mathematical Formulas and Integrals, Elsevier Academic Press, San Diego, Calif, USA, 3rd edition, 2004. View at: MathSciNet
31. D. S. Mitrinović, Analytic Inequalities, Springer, New York, NY, USA, 1970. View at: MathSciNet
32. J. C. Kuang, Applied Inequalities, Shangdong Science and Technology Press, Jinan, China, 3rd edition, 2004.

#### More related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.