- Research Article
- Open Access

# Inequalities for Hyperbolic Functions and Their Applications

- L Zhu
^{1}Email author

**2010**:130821

https://doi.org/10.1155/2010/130821

© L. Zhu. 2010

**Received:**9 April 2010**Accepted:**15 May 2010**Published:**14 June 2010

## Abstract

A basic theorem is established and found to be a source of inequalities for hyperbolic functions, such as the ones of Cusa, Huygens, Wilker, Sandor-Bencze, Carlson, Shafer-Fink type inequality, and the one in the form of Oppenheim's problem. Furthermore, these inequalities described above will be extended by this basic theorem.

## Keywords

- Type Inequality
- Circular Function
- Exponential Type
- Infinite Series
- Hyperbolic Function

## 1. Introduction

In the study by Zhu in [1], a basic theorem is established and found to be a source of inequalities for circular functions, and these inequalities are extended by this basic theorem. In what follows we are going to present the counterpart of these results for the hyperbolic functions.

In this paper, we first establish the following Cusa-type inequalities in exponential type for hyperbolic functions described as Theorem 1.1. Then using the results of Theorem 1.1, we obtain Huygens, Wilker, Sandor-Bencze, Carlson, and Shafer-Fink-type inequalities in Sections 4, 5, 6, 7, 8, respectively.

Theorem 1.1 (Cusa-type inequalities).

Let . Then the following are considered.

holds if and only if and

holds if and only if .

holds if and only if .

## 2. Lemmas

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.

hold.

Proof.

## 3. Proof of Theorem 1

We obtain results in the following two cases.

So and is decreasing on . This leads to that is decreasing on by Lemma 2.1. At the same time, using power series expansions, we have that , and rewriting as , we see that . So the proof of (i) in Theorem 1.1 is complete.

(b)When , by (3.2), (2.2), and (2.1) we obtain

So and is increasing on and the function is increasing on by Lemma 2.1. At the same time, , but . So the proof of (ii) in Theorem 1.1 is complete.

## 4. Huygens-Type Inequalities

Multiplying three functions by showed in (1.1) and (1.2), we can obtain the following results on Huygens-type inequalities for the hyperbolic functions.

Theorem 4.1.

Let . Then one has the following.

holds if and only if and .

holds if and only if .

and holds if and only if .

When letting in (4.1) and in (4.3), one can obtain two results of Zhu [19].

Corollary 4.2 (see [19, Theorem ]).

holds for all if and only if and .

Corollary 4.3 (see [19, Theorem ]).

holds for all if and only if .

When letting in (4.4), one can obtain a result on Cusa-type inequality (see the study by Baricz and Zhu in [20]).

Corollary 4.4 (see [20, Theorem ]).

holds for all .

When letting in (4.5), one can obtain a new result on Huygens-type inequality.

Corollary 4.5.

holds for all if and only if .

Remark 4.6.

## 5. Wilker-Type Inequalities

In this section, we obtain the following results on Wilker-type inequalities.

Theorem 5.1.

Let . Then the following are considered.

holds.

holds.

by the arithmetic mean-geometric mean inequality and the right of inequality (4.1). By (5.1), we have (5.2).

One can obtain the following three results from Theorem 5.1.

Corollary 5.2 (First Wilker-type inequality, see [24]).

holds for all .

Corollary 5.3 (Second Wilker-type inequality).

holds for all .

Corollary 5.4.

holds for all .

Remark 5.5.

Inequality (5.2) is a generalization of a result of Zhu [22] since (5.2) holds for while it holds for in [22].

## 6. Sandor-Bencze-Type Inequalities

From Theorem 1.1, we can obtain some results on Sandor-Bencze-type inequalities (Sandor-Bencze inequalities for circular functions can be found in [25]).

Theorem 6.1.

Let . Then the following are considered.

## 7. Carlson-Type Inequalities

Replacing with and letting in Theorem 1.1, we have the following.

Theorem 7.1.

Let . Then the following are considered.

holds.

(2)When , the left inequality of (7.2) holds too.

When letting in Theorem 7.1, one can obtain the following result.

Corollary 7.2.

holds.

## 8. Shafer-Fink-Type Inequalities and an Extension of the Problem of Oppenheim

First, let and in Theorem 1.1, then , and we have the following.

Theorem 8.1.

holds.

Theorem 8.1 can deduce to the following result.

Corollary 8.2 (see [26]).

Second, let for and in Theorem 1.1. Then let or . Since and , one obtains the following result.

Theorem 8.3.

holds.

Theorem 8.3 can deduce to the following result.

Corollary 8.4 (see [26]).

Finally, Theorem 1.1 is equivalent to the following statement which modifies a problem of Oppenheim (a problem of Oppenheim for circular functions can be found in [20, 27–29]).

Theorem 8.5.

holds.

## Authors’ Affiliations

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