Research Article | Open Access

# 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.

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#### 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.