# On Jordan Type Inequalities for Hyperbolic Functions

- R Klén
^{1}Email author, - M Visuri
^{1}and - M Vuorinen
^{1}

**2010**:362548

https://doi.org/10.1155/2010/362548

© R. Klén et al. 2010

**Received: **28 January 2010

**Accepted: **29 April 2010

**Published: **31 May 2010

## Abstract

## Keywords

## 1. Introduction

has been in the focus of these studies and many refinements have been proved for it by Wu and Srivastava [9, 10], Zhang et al. [11], J.-L. Li and Y.-L. Li [5, 12], Wu and Debnath [13–15], Özban [16], Qi et al. [17], Zhu [18–29], Sándor [30, 31], Baricz and Wu [32, 33], Neuman and Sándor [34], Agarwal et al. [35], Niu et al. [36], Pan and Zhu [37], and Qi and Guo [38]. For a long list of recent papers on this topic see [7] and for an extensive survey see [17]. The proofs are based on familiar methods of calculus. In particular, a method based on a l'Hospital type criterion for monotonicity of the quotient of two functions from Anderson et al. [39] is a key tool in these studies. Some other applications of this criterion are reviewed in [40, 41]. Pinelis has found several applications of this criterion in [42] and in several other papers.

where is well-known and it was studied recently by Baricz in [43, page 111]. The second inequality of (1.2) is given in [8, page 354, 3.9.32] for For a refinement of the first inequality in (1.2) see Remark 1.3(1) and of the second inequality see Theorem 2.4.

This paper is motivated by these studies and it is based on the Master Thesis of Visuri [44]. Some of our main results are the following theorems.

Theorem 1.1.

Theorem 1.2.

We will consider quotients and at origin as limiting values and .

## 2. Jordan's Inequality

In this section we will find upper and lower bounds for by using hyperbolic trigonometric functions.

Theorem 2.1.

Proof.

and the function is increasing on . Now for .

Proof of Theorem 1.1.

The upper bound of is clear by Theorem 2.1. The lower bound of holds true if the function is positive on .

is positive which implies the assertion.

Let us denote . Now and therefore and . Therefore inequality (2.8) holds for and the assertion follows.

We next show that for the upper and lower bounds of (1.2) are better than the upper and lower bounds in Theorem 2.1.

- (i)

and for all . Now is a convex function on and is nondecreasing on with . Therefore is nondecreasing and .

- (ii)

which implies the first inequality of the claim. The second inequality is trivial since .

Theorem 2.3.

(iii)the functions and are decreasing on .

and is equivalent to . Since we have . Therefore is increasing on .

and . Therefore the function is increasing on and is decreasing on .

and . Therefore the function is increasing on and is decreasing on .

We next will improve the upper bound of (1.2).

Theorem 2.4.

Proof.

The first inequality of (2.24) follows from (1.2).

where the second inequality is equivalent to and the second inequality of (2.24) follows.

and by the Leibniz Criterion the assertion follows.

## 3. Hyperbolic Jordan's Inequality

In this section we will find upper and lower bounds for the functions and .

Theorem 3.1.

Proof.

which proves the lower bound.

and the assertion follows from inequality .

Remark 3.2.

where This result improves Theorem 3.1.

Lemma 3.3.

Proof of Theorem 1.2.

and the assertion follows.

Theorem 3.4.

Proof.

Therefore and the assertion follows.

Theorem 3.5.

Proof.

The upper bound of holds true if the function is positive on . Since the function is increasing. Therefore and .

and the assertion follows.

Remark 3.6.

Baricz and Wu have shown in [33, page 276-277] that the right hand side of Theorem 2.1 is true for and the right hand side of Theorem 3.5 is true for . Their proof is based on the infinite product representations.

Hence, the upper bound in Theorem 3.5 is better that in Theorem 3.4.

## 4. Trigonometric Inequalities

Theorem 4.1.

For the following inequalities hold

which is true because is decreasing on and .

and since on the assertion follows.

Remark 4.2.

Similar inequalities to Theorem 4.1 have been considered by Neuman in [46, page 34-35].

Theorem 4.3.

## Authors’ Affiliations

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