#### 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.