Some New Wilker-Type Inequalities for Circular and Hyperbolic Functions
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.
Wilker  proposed two open questions, the first of which was the following statement.
Problem 1. Let . Then holds.
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.
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 2.4. Let , then the inequality holds.
Proof. The following power series expansion can be found in [30, 188.8.131.52 ( 3)] Then
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.
Proof of Lemma 2.9. Let , where , and . Then
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
Combining (4.5) and (4.7) gives (1.6).
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.
7. Open Problem
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