Inequalities for Hyperbolic Functions and Their Applications

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.

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.

(i)If , the double inequality

(1.1)

holds if and only if and

(ii)If , the inequality

(1.2)

holds if and only if .

That is, let , then the inequality

(1.3)

holds if and only if .

2. Lemmas

Lemma 2.1 (see [2–18]).

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.

Let . Then the inequalities

(2.1)

(2.2)

(2.3)

hold.

Proof.

Using the infinite series of , and , we have

(2.4)

(2.5)

(2.6)

3. Proof of Theorem  1

Let , where , and . Then

(3.1)

where

(3.2)

We obtain results in the following two cases.

(a)When , by (3.2), (2.2), and (2.3) we have

(3.3)

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

(3.4)

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.

(1)When , the double inequality

(4.1)

holds if and only if and .

(2)When , the inequality

(4.2)

holds if and only if .

Let , then inequality (4.2) is equivalent to

(4.3)

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 ]).

One has that

(4.4)

holds for all if and only if and .

Corollary 4.3 (see [19, Theorem ]).

One has that

(4.5)

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 ]).

One has that

(4.6)

or

(4.7)

that is,

(4.8)

holds for all .

Inequality (4.6) can deduce to the following one which is from the study by Baricz in [21]:

(4.9)

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

Corollary 4.5.

One has that

(4.10)

holds for all if and only if .

Remark 4.6.

Attention is drawn to the fact that, comparing Cusa-type inequality with Huygens-type inequality, Neuman and Sandor [21] obtained the following result:

(4.11)

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.

(i)When , the inequality

(5.1)

holds.

(ii)When , then the inequality

(5.2)

holds.

Proof.

• (i) The proof of (i) can be seen in [22, 23].

• (ii) When , we can obtain

  

  (5.3)

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]).

One has that

(5.4)

holds for all .

Corollary 5.3 (Second Wilker-type inequality).

One has that

(5.5)

holds for all .

Corollary 5.4.

One has that

(5.6)

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.

(1)When , one has

(6.1)

(2)When , one has

(6.2)

7. Carlson-Type Inequalities

Let for , then for , and

(7.1)

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

Theorem 7.1.

Let . Then the following are considered.

(1)When , the double inequality

(7.2)

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.

Let . Then the double inequality

(7.3)

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.

Let , or . Then the inequality

(8.1)

holds.

Theorem 8.1 can deduce to the following result.

Corollary 8.2 (see [26]).

Let . Then

(8.2)

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

Theorem 8.3.

Let , or . Then the inequality

(8.3)

holds.

Theorem 8.3 can deduce to the following result.

Corollary 8.4 (see [26]).

Let . Then

(8.4)

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.

Let or . Then the inequality

(8.5)

holds.