- Research Article
- Open Access

# 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

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.

## Keywords

- Series Expansion
- Simple Computation
- Straightforward Computation
- Type Inequality
- Master Thesis

## 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 .

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 .

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.

- (iii)
For

- (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.

Let . Then

is increasing on ,

is decreasing on ,

(iii)the functions and are decreasing on .

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

- (iii)

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.

For

(i)

(ii)

(iii)

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

(i)

(ii)

(iii)

(iv)

which is true because is decreasing on and .

- (iii)

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

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)
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.

the assertion follows from the assumptions and .

## Authors’ Affiliations

## References

- Baricz Á: Redheffer type inequality for Bessel functions.
*Journal of Inequalities in Pure and Applied Mathematics*2007, 8(1, article 11):1–6.MathSciNetMATHGoogle Scholar - Baricz Á: Jordan-type inequalities for generalized Bessel functions.
*Journal of Inequalities in Pure and Applied Mathematics*2008, 9(2, article 39):1–6.MathSciNetMATHGoogle Scholar - Debnath L, Zhao C-J: New strengthened Jordan's inequality and its applications.
*Applied Mathematics Letters*2003, 16(4):557–560. 10.1016/S0893-9659(03)00036-3MathSciNetView ArticleMATHGoogle Scholar - Jiang WD, Yun H: Sharpening of Jordan's inequality and its applications.
*Journal of Inequalities in Pure and Applied Mathematics*2006, 7(3, article 102):1–4.MathSciNetMATHGoogle Scholar - Li J-L: An identity related to Jordan's inequality.
*International Journal of Mathematics and Mathematical Sciences*2006, 2006:-6.Google Scholar - Wu S-H, Srivastava HM, Debnath L: Some refined families of Jordan-type inequalities and their applications.
*Integral Transforms and Special Functions*2008, 19(3–4):183–193.MathSciNetView ArticleMATHGoogle Scholar - Zhu L, Sun J: Six new Redheffer-type inequalities for circular and hyperbolic functions.
*Computers & Mathematics with Applications*2008, 56(2):522–529. 10.1016/j.camwa.2008.01.012MathSciNetView ArticleMATHGoogle Scholar - Mitrinović DS:
*Analytic Inequalities, Die Grundlehren der mathematischen Wissenschaften*.*Volume 16*. Springer, New York, NY, USA; 1970:xii+400. in cooperation with P. M. VasićGoogle Scholar - Wu S-H, Srivastava HM: A weighted and exponential generalization of Wilker's inequality and its applications.
*Integral Transforms and Special Functions*2007, 18(7–8):529–535.MathSciNetView ArticleMATHGoogle Scholar - Wu S-H, Srivastava HM: A further refinement of a Jordan type inequality and its application.
*Applied Mathematics and Computation*2008, 197(2):914–923. 10.1016/j.amc.2007.08.022MathSciNetView ArticleMATHGoogle Scholar - Zhang X, Wang G, Chu Y: Extensions and sharpenings of Jordan's and Kober's inequalities.
*Journal of Inequalities in Pure and Applied Mathematics*2006, 7(2, article 63):1–3.MathSciNetMATHGoogle Scholar - Li J-L, Li Y-L: On the strengthened Jordan's inequality.
*Journal of Inequalities and Applications*2007, 2007:-8.Google Scholar - Wu S, Debnath L: A new generalized and sharp version of Jordan's inequality and its applications to the improvement of the Yang Le inequality.
*Applied Mathematics Letters*2006, 19(12):1378–1384. 10.1016/j.aml.2006.02.005MathSciNetView ArticleMATHGoogle Scholar - Wu S, Debnath L: A new generalized and sharp version of Jordan's inequality and its applications to the improvement of the Yang Le inequality. II.
*Applied Mathematics Letters*2007, 20(5):532–538. 10.1016/j.aml.2006.05.022MathSciNetView ArticleMATHGoogle Scholar - Wu S, Debnath L: Jordan-type inequalities for differentiable functions and their applications.
*Applied Mathematics Letters*2008, 21(8):803–809. 10.1016/j.aml.2007.09.001MathSciNetView ArticleMATHGoogle Scholar - Özban AY: A new refined form of Jordan's inequality and its applications.
*Applied Mathematics Letters*2006, 19(2):155–160. 10.1016/j.aml.2005.05.003MathSciNetView ArticleMATHGoogle Scholar - Qi F, Niu D-W, Guo B-N: Refinements, generalizations, and applications of Jordan's inequality and related problems.
*Journal of Inequalities and Applications*2009, 2009:-52.Google Scholar - Zhu L: Sharpening of Jordan's inequalities and its applications.
*Mathematical Inequalities & Applications*2006, 9(1):103–106.MathSciNetView ArticleMATHGoogle Scholar - Zhu L: Sharpening Jordan's inequality and the Yang Le inequality.
*Applied Mathematics Letters*2006, 19(3):240–243. 10.1016/j.aml.2005.06.004MathSciNetView ArticleMATHGoogle Scholar - Zhu L: Sharpening Jordan's inequality and Yang Le inequality. II.
*Applied Mathematics Letters*2006, 19(9):990–994. 10.1016/j.aml.2005.11.011MathSciNetView ArticleMATHGoogle Scholar - Zhu L: A general refinement of Jordan-type inequality.
*Computers & Mathematics with Applications*2008, 55(11):2498–2505. 10.1016/j.camwa.2007.10.004MathSciNetView ArticleMATHGoogle Scholar - Zhu L: General forms of Jordan and Yang Le inequalities.
*Applied Mathematics Letters*2009, 22(2):236–241. 10.1016/j.aml.2008.03.017MathSciNetView ArticleMATHGoogle Scholar - Zhu L: Sharpening Redheffer-type inequalities for circular functions.
*Applied Mathematics Letters*2009, 22(5):743–748. 10.1016/j.aml.2008.08.012MathSciNetView ArticleMATHGoogle Scholar - Zhu L: Some new inequalities of the Huygens type.
*Computers & Mathematics with Applications*2009, 58(6):1180–1182. 10.1016/j.camwa.2009.07.045MathSciNetView ArticleMATHGoogle Scholar - Zhu L: Some new Wilker-type inequalities for circular and hyperbolic functions.
*Abstract and Applied Analysis*2009, 2009:-9.Google Scholar - Zhu L: A source of inequalities for circular functions.
*Computers & Mathematics with Applications*2009, 58(10):1998–2004. 10.1016/j.camwa.2009.07.076MathSciNetView ArticleMATHGoogle Scholar - Zhu L: Generalized lazarevic's inequality and its applications—part II.
*Journal of Inequalities and Applications*2009, 2009:-4.Google Scholar - Zhu L: Jordan type inequalities involving the Bessel and modified Bessel functions.
*Computers & Mathematics with Applications*2010, 59(2):724–736. 10.1016/j.camwa.2009.10.020MathSciNetView ArticleMATHGoogle Scholar - Zhu L: Inequalities for hyperbolic functions and their applications. Journal of Inequalities and Applications. In pressGoogle Scholar
- Sándor J: On the concavity of .
*Octogon Mathematical Magazine*2005, 13(1):406–407.Google Scholar - Sándor J: A note on certain Jordan type inequalities. RGMIA Research Report Collection 2007., 10(1, article 1):Google Scholar
- Baricz Á, Wu S: Sharp Jordan-type inequalities for Bessel functions.
*Publicationes Mathematicae Debrecen*2009, 74(1–2):107–126.MathSciNetMATHGoogle Scholar - Baricz Á, Wu S: Sharp exponential Redheffer-type inequalities for Bessel functions.
*Publicationes Mathematicae Debrecen*2009, 74(3–4):257–278.MathSciNetMATHGoogle Scholar - Neuman E, Sándor J: On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities.
*Mathematical Inequalities & Applications*2010, 1973: 1–9.MathSciNetMATHGoogle Scholar - Agarwal RP, Kim Y-H, Sen SK: A new refined Jordan's inequality and its application.
*Mathematical Inequalities & Applications*2009, 12(2):255–264.MathSciNetView ArticleMATHGoogle Scholar - Niu D-W, Huo Z-H, Cao J, Qi F: A general refinement of Jordan's inequality and a refinement of L. Yang's inequality.
*Integral Transforms and Special Functions*2008, 19(3–4):157–164.MathSciNetView ArticleMATHGoogle Scholar - Pan W, Zhu L: Generalizations of Shafer-Fink-type inequalities for the arc sine function.
*Journal of Inequalities and Applications*2009, 2009:-6.Google Scholar - Qi F, Guo B-N: A concise proof of Oppenheim's double inequality relating to the cosine and sine functions. http://arxiv.org/abs/arxiv:0902.2511
- Anderson GD, Vamanamurthy MK, Vuorine M:
*Conformal Invariants, Inequalities and Quasiconformal Mappings*. John Wiley & Sons, New York, NY, USA; 1997.Google Scholar - Anderson GD, Vamanamurthy MK, Vuorinen M: Topics in special functions. In
*Papers on Analysis: A Volume Dedicated to Olli Martio on the Occasion of His 60th Birthday, Rep. Univ. Jyväskylä Dep. Math. Stat.*.*Volume 83*. Edited by: Heinonen J, Kilpeläinen T, Koskela P. Univ. Jyväskylä, Jyväskylä, Finland; 2001:5–26.Google Scholar - Anderson GD, Vamanamurthy MK, Vuorinen M: Topics in special functions. II.
*Conformal Geometry and Dynamics*2007, 11: 250–270. 10.1090/S1088-4173-07-00168-3MathSciNetView ArticleMATHGoogle Scholar - Pinelis I: l'Hospital rules for monotonicity and the Wilker-Anglesio inequality.
*The American Mathematical Monthly*2004, 111(10):905–909. 10.2307/4145099MathSciNetView ArticleMATHGoogle Scholar - Baricz Á:
*Generalized Bessel functions of the first kind, Ph.D. thesis*. Babes-Bolyai University, Cluj-Napoca, Romania; 2008.Google Scholar - Lehtonen M:
*Yleistetty konveksisuus, M.S. thesis*. University of Turku; March 2008. written under the supervision of Prof. M. VuorinenGoogle Scholar - Qi F, Cui L-H, Xu S-L: Some inequalities constructed by Tchebysheff's integral inequality.
*Mathematical Inequalities & Applications*1999, 2(4):517–528.MathSciNetView ArticleMATHGoogle Scholar - Neuman E: Inequalities involving inverse circular and inverse hyperbolic functions.
*Publikacije Elektrotehničkog Fakulteta Univerzitet u Beogradu. Serija Matematika*2007, 18: 32–37.MathSciNetView ArticleMATHGoogle Scholar

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