On Jordan Type Inequalities for Hyperbolic Functions

Abstract

This paper deals with some inequalities for trigonometric and hyperbolic functions such as the Jordan inequality and its generalizations. In particular, lower and upper bounds for functions such as and are proved.

1. Introduction

During the past several years there has been a great deal of interest in trigonometric inequalities [1–7]. The classical Jordan inequality [8, page 31]

(1.1)

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.

The inequality

(1.2)

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.

For

(1.3)

Theorem 1.2.

For

(1.4)

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

Remark 1.3.

• (1) Let

  

  (1.5)

Then on because

(1.6)

In [45, ()] it is proved that for

(1.7)

Hence is a better lower bound for than (1.2) for .

• (2) Observe that

  

  (1.8)

which holds true as equality if and only if . In conclusion, (1.8) holds for all . Together with (1.2) we now have

(1.9)

and by (1.8)

(1.10)

2. Jordan's Inequality

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

Theorem 2.1.

For

(2.1)

Proof.

The lower bound of holds true if the function is positive on . Since

(2.2)

we have for and is increasing on . Therefore

(2.3)

and the function is increasing on . Now for .

The upper bound of holds true if the function is positive on . Let us denote . Since for we have and for . Now

(2.4)

which is positive on , because and for . Therefore

(2.5)

for . 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 .

Let us assume . Since we have . We will show that

(2.6)

is positive which implies the assertion.

Now is equivalent to

(2.7)

Since it is sufficient to show that , which is equivalent to

(2.8)

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.

Theorem 2.2.

• (i) For

  

  (2.9)

• (ii) For

  

  (2.10)

• (iii) For

(2.11)

Proof.

• (i) The claim holds true if the function is nonnegative on . By a simple computation we obtain . Inequality is equivalent to . By the series expansions of and we obtain

  

  (2.12)

where is the th Bernoulli number. By the properties of the Bernoullin numbers , , coefficients , for , form an alternating sequence, as and for . Therefore by Leibniz Criterion

(2.13)

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

• (ii) The claim holds true if the function is nonnegative on . By the series expansion of we have and therefore by the series expansion of

  

  (2.14)

and the assertion follows.

• (iii) Clearly we have

  

  (2.15)

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

Theorem 2.3.

Let . Then

(i)the function

(2.16)

is increasing on ,

(ii)the function

(2.17)

is decreasing on ,

(iii)the functions and are decreasing on .

Proof.

• (i) Let us consider instead of the function

  

  (2.18)

for . Note that and therefore the claim is equivalent to the function being decreasing on . We have

(2.19)

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

• (ii) We will consider instead of the function

  

  (2.20)

for . Note that and therefore the claim is equivalent to the function being increasing on . We have

(2.21)

and is equivalent to . Since we have . Therefore and the assertion follows.

• (iii) We will show that is increasing on . Now ,

  

  (2.22)

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

We will show that is increasing on . Now ,

(2.23)

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

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

Theorem 2.4.

For

(2.24)

Proof.

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

By the series expansions of and

(2.25)

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

By the identity the upper bound of (2.24) is equivalent to . By the series expansion of

(2.26)

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.

For

(3.1)

Proof.

We obtain from the series expansion of

(3.2)

which proves the lower bound.

By using the identity the chain of inequalities (1.2) gives

(3.3)

and the assertion follows from inequality .

Remark 3.2.

J.-L. Li and Y.-L. Li have proved [12, (4.9)] that

(3.4)

where This result improves Theorem 3.1.

Lemma 3.3.

For

(i)

(ii)

(iii)

Proof.

• (i) For we have which is equivalent to

  

  (3.5)

By Theorems 2.1, 3.1, and (3.5)

(3.6)

• (ii) Since for we have

  

  (3.7)

• (iii) By the series expansion of we have

  

  (3.8)

Proof of Theorem 1.2.

The lower bound of follows from Lemma 3.3 and Theorem 3.1 since

(3.9)

The upper bound of holds true if the function is positive on . By the series expansion it is clear that

(3.10)

By Lemma 3.3 and (3.10)

(3.11)

and the assertion follows.

Theorem 3.4.

For

(3.12)

Proof.

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

(3.13)

we have

(3.14)

Therefore and the assertion follows.

Theorem 3.5.

For

(3.15)

Proof.

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

The lower bound of holds true if the function is positive on . By the series expansions we have

(3.16)

By a straightforward computation we see that the polynomial is strictly decreasing on . Therefore

(3.17)

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.

Note that for

(3.18)

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

(i)

(ii)

(iii)

(iv)

Proof.

• (i) By setting the assertion is equivalent to

  

  (4.1)

which is true because is decreasing on and .

• (ii) By the series expansions of and we have by Leibniz Criterion

  

  (4.2)

and since on the assertion follows.

• (iii) By the series expansions of and we have by Leibniz Criterion

  

  (4.3)

and since on the assertion follows.

• (iv) By the series expansions of and we have by Leibniz Criterion

(4.4)

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.

Let . Then

(i)for

(4.5)

(ii)for

(4.6)

(iii)for

(4.7)

Proof.

• (i) The claim follows from the fact that is decreasing on .

• (ii) The claim is equivalent to saying that the function is increasing for . Since and is equivalent to the assertion follows.

(iii)The claim is equivalent to . By the series expansion of we have

(4.8)

where is the th Bernoulli number (, , ). The assertion follows from the Leibniz Criterion, if

(4.9)

for all . Since (4.9) is equivalent to

(4.10)

the assertion follows from the assumptions and .