Open Access

On Jordan Type Inequalities for Hyperbolic Functions

Journal of Inequalities and Applications20102010:362548

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

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.

1. Introduction

During the past several years there has been a great deal of interest in trigonometric inequalities [17]. 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 [1315], Özban [16], Qi et al. [17], Zhu [1829], 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. (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 .
  1. (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.
  1. (i)
    For
    (2.9)
     
  1. (ii)
    For
    (2.10)
     
  1. (iii)

    For

     
(2.11)
Proof.
  1. (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 .

  1. (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.
  1. (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.
  1. (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 .

  1. (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.
  1. (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.
  1. (i)
    For we have which is equivalent to
    (3.5)
     
By Theorems 2.1, 3.1, and (3.5)
(3.6)
  1. (ii)
    Since for we have
    (3.7)
     
  1. (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.
  1. (i)
    By setting the assertion is equivalent to
    (4.1)
     

which is true because is decreasing on and .

  1. (ii)
    By the series expansions of and we have by Leibniz Criterion
    (4.2)
     
and since on the assertion follows.
  1. (iii)
    By the series expansions of and we have by Leibniz Criterion
    (4.3)
     
and since on the assertion follows.
  1. (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.
  1. (i)

    The claim follows from the fact that is decreasing on .

     
  1. (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 .

Authors’ Affiliations

(1)
Department of Mathematics, University of Turku

References

  1. 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
  2. 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
  3. 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
  4. 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
  5. Li J-L: An identity related to Jordan's inequality. International Journal of Mathematics and Mathematical Sciences 2006, 2006:-6.Google Scholar
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. Li J-L, Li Y-L: On the strengthened Jordan's inequality. Journal of Inequalities and Applications 2007, 2007:-8.Google Scholar
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. Zhu L: Sharpening of Jordan's inequalities and its applications. Mathematical Inequalities & Applications 2006, 9(1):103–106.MathSciNetView ArticleMATHGoogle Scholar
  19. 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
  20. 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
  21. 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
  22. 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
  23. 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
  24. 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
  25. Zhu L: Some new Wilker-type inequalities for circular and hyperbolic functions. Abstract and Applied Analysis 2009, 2009:-9.Google Scholar
  26. 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
  27. Zhu L: Generalized lazarevic's inequality and its applications—part II. Journal of Inequalities and Applications 2009, 2009:-4.Google Scholar
  28. 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
  29. Zhu L: Inequalities for hyperbolic functions and their applications. Journal of Inequalities and Applications. In pressGoogle Scholar
  30. Sándor J: On the concavity of . Octogon Mathematical Magazine 2005, 13(1):406–407.Google Scholar
  31. Sándor J: A note on certain Jordan type inequalities. RGMIA Research Report Collection 2007., 10(1, article 1):Google Scholar
  32. Baricz Á, Wu S: Sharp Jordan-type inequalities for Bessel functions. Publicationes Mathematicae Debrecen 2009, 74(1–2):107–126.MathSciNetMATHGoogle Scholar
  33. Baricz Á, Wu S: Sharp exponential Redheffer-type inequalities for Bessel functions. Publicationes Mathematicae Debrecen 2009, 74(3–4):257–278.MathSciNetMATHGoogle Scholar
  34. 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
  35. 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
  36. 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
  37. 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
  38. 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
  39. Anderson GD, Vamanamurthy MK, Vuorine M: Conformal Invariants, Inequalities and Quasiconformal Mappings. John Wiley & Sons, New York, NY, USA; 1997.Google Scholar
  40. 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
  41. 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
  42. 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
  43. Baricz Á: Generalized Bessel functions of the first kind, Ph.D. thesis. Babes-Bolyai University, Cluj-Napoca, Romania; 2008.Google Scholar
  44. Lehtonen M: Yleistetty konveksisuus, M.S. thesis. University of Turku; March 2008. written under the supervision of Prof. M. VuorinenGoogle Scholar
  45. 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
  46. 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

Copyright

© R. Klén et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.