Abstract and Applied Analysis

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.