Research Article | Open Access

Ling Zhu, "Some New Wilker-Type Inequalities for Circular and Hyperbolic Functions", *Abstract and Applied Analysis*, vol. 2009, Article ID 485842, 9 pages, 2009. https://doi.org/10.1155/2009/485842

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

**Academic Editor:**Ferhan Atici

#### 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 [1] proposed two open questions, the first of which was the following statement.

*Problem 1. *Let . Then
holds.

Sumner et al. [2] proved inequality (1.1). Guo et al. [3] 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 [6] obtained Wilker-type inequality as follows: Baricz and Sandor [7] 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 [5] 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 [8–24]). *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 [25–27]). *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.

#### References

- J. B. Wilker, “E3306,”
*The American Mathematical Monthly*, vol. 96, no. 1, p. 55, 1989. View at: Google Scholar - 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 - 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. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. Zhu, “A new simple proof of Wilker's inequality,”
*Mathematical Inequalities & Applications*, vol. 8, no. 4, pp. 749–750, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. Zhu, “On Wilker-type inequalities,”
*Mathematical Inequalities & Applications*, vol. 10, no. 4, pp. 727–731, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet - 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 - 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. View at: Google Scholar | Zentralblatt MATH | MathSciNet - 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. View at: Google Scholar | Zentralblatt MATH | MathSciNet - 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. View at: Google Scholar | Zentralblatt MATH | MathSciNet - 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 - L. Zhu, “Sharpening Jordan's inequality and the Yang Le inequality,”
*Applied Mathematics Letters*, vol. 19, no. 3, pp. 240–243, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. Zhu, “Sharpening Jordan's inequality and Yang Le inequality. II,”
*Applied Mathematics Letters*, vol. 19, no. 9, pp. 990–994, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. Zhu, “Sharpening of Jordan's inequalities and its applications,”
*Mathematical Inequalities & Applications*, vol. 9, no. 1, pp. 103–106, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet - 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 - L. Zhu, “A general refinement of Jordan-type inequality,”
*Computers & Mathematics with Applications*, vol. 55, no. 11, pp. 2498–2505, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - 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 - 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. View at: Google Scholar | Zentralblatt MATH | MathSciNet - 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 - 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 - 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 - 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 - 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 - S. Ponnusamy and M. Vuorinen, “Asymptotic expansions and inequalities for hypergeometric functions,”
*Mathematika*, vol. 44, no. 2, pp. 278–301, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet - 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 - A. Jeffrey,
*Handbook of Mathematical Formulas and Integrals*, Elsevier Academic Press, San Diego, Calif, USA, 3rd edition, 2004. View at: MathSciNet - D. S. Mitrinović,
*Analytic Inequalities*, Springer, New York, NY, USA, 1970. View at: MathSciNet - J. C. Kuang,
*Applied Inequalities*, Shangdong Science and Technology Press, Jinan, China, 3rd edition, 2004.

#### Copyright

Copyright © 2009 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.