Open Access

Inequalities for Hyperbolic Functions and Their Applications

Journal of Inequalities and Applications20102010:130821

https://doi.org/10.1155/2010/130821

Received: 9 April 2010

Accepted: 15 May 2010

Published: 14 June 2010

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 [218]).

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.
  1. (i)

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

     
  2. (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, 2729]).

Theorem 8.5.

Let or . Then the inequality
(8.5)

holds.

Authors’ Affiliations

(1)
Department of Mathematics, Zhejiang Gongshang University

References

  1. Zhu L: A source of inequalities for circular functions. Computers & Mathematics with Applications 2009, 58(10):1998–2004. 10.1016/j.camwa.2009.07.076MATHMathSciNetView ArticleGoogle Scholar
  2. Anderson GD, Vamanamurthy MK, Vuorinen M: Inequalities for quasiconformal mappings in space. Pacific Journal of Mathematics 1993, 160(1):1–18.MATHMathSciNetView ArticleGoogle Scholar
  3. Anderson GD, Qiu S-L, Vamanamurthy MK, Vuorinen M: Generalized elliptic integrals and modular equations. Pacific Journal of Mathematics 2000, 192(1):1–37. 10.2140/pjm.2000.192.1MathSciNetView ArticleGoogle Scholar
  4. Pinelis I: L'Hospital type results for monotonicity, with applications. Journal of Inequalities in Pure and Applied Mathematics 2002, 3(1, article 5):1–5.Google Scholar
  5. Pinelis I: "Non-strict" l'Hospital-type rules for monotonicity: intervals of constancy. Journal of Inequalities in Pure and Applied Mathematics 2007, 8(1, article 14):1–8.MathSciNetGoogle Scholar
  6. Zhu L: Sharpening Jordan's inequality and the Yang Le inequality. Applied Mathematics Letters 2006, 19(3):240–243. 10.1016/j.aml.2005.06.004MATHMathSciNetView ArticleGoogle Scholar
  7. Zhu L: Sharpening Jordan's inequality and Yang Le inequality. II. Applied Mathematics Letters 2006, 19(9):990–994. 10.1016/j.aml.2005.11.011MATHMathSciNetView ArticleGoogle Scholar
  8. Zhu L: Sharpening of Jordan's inequalities and its applications. Mathematical Inequalities & Applications 2006, 9(1):103–106.MATHMathSciNetView ArticleGoogle Scholar
  9. Zhu L: Some improvements and generalizations of Jordan's inequality and Yang Le inequality. In Inequalities and Applications. Edited by: Rassias ThM, Andrica D. CLUJ University Press, Cluj-Napoca, Romania; 2008.Google Scholar
  10. Zhu L: A general refinement of Jordan-type inequality. Computers & Mathematics with Applications 2008, 55(11):2498–2505. 10.1016/j.camwa.2007.10.004MATHMathSciNetView ArticleGoogle Scholar
  11. 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
  12. 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.2006.02.005MATHMathSciNetView ArticleGoogle 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, II. Applied Mathematics Letters 2007, 20(5):532–538. 10.1016/j.aml.2006.05.022MATHMathSciNetView ArticleGoogle Scholar
  14. 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.MATHMathSciNetView ArticleGoogle Scholar
  15. 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.MATHMathSciNetView ArticleGoogle Scholar
  16. Zhu L: General forms of Jordan and Yang Le inequalities. Applied Mathematical Letters 2009, 22(2):236–241. 10.1016/j.aml.2008.03.017MATHView ArticleGoogle Scholar
  17. 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.2007.09.001MATHMathSciNetView ArticleGoogle Scholar
  18. Wu S, Debnath L: A generalization of L'Hôspital-type rules for monotonicity and its application. Applied Mathematics Letters 2009, 22(2):284–290. 10.1016/j.aml.2008.06.001MATHMathSciNetView ArticleGoogle Scholar
  19. Zhu L: Some new inequalities of the Huygens type. Computers & Mathematics with Applications 2009, 58(6):1180–1182. 10.1016/j.camwa.2009.07.045MATHMathSciNetView ArticleGoogle Scholar
  20. Baricz Á, Zhu L: Extension of Oppenheim's problem to Bessel functions. Journal of Inequalities and Applications 2007, 2007:-7.Google Scholar
  21. Neuman E, Sandor J: On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa–Huygens, Wilker, and Huygens inequalities. Mathematical Inequalities & Applications, Preprint, 2010 Mathematical Inequalities & Applications, Preprint, 2010Google Scholar
  22. Zhu L: Some new Wilker-type inequalities for circular and hyperbolic functions. Abstract and Applied Analysis 2009, 2009:-9.Google Scholar
  23. Wu S, Baricz Á: Generalizations of Mitrinović, Adamović and Lazarević's inequalities and their applications. Publicationes Mathematicae 2009, 75(3–4):447–458.MATHMathSciNetGoogle Scholar
  24. Zhu L: On Wilker-type inequalities. Mathematical Inequalities & Applications 2007, 10(4):727–731.MATHMathSciNetView ArticleGoogle Scholar
  25. Baricz Á, Sándor J: Extensions of the generalized Wilker inequality to Bessel functions. Journal of Mathematical Inequalities 2008, 2(3):397–406.MATHMathSciNetView ArticleGoogle Scholar
  26. Zhu L: New inequalities of Shafer-Fink type for arc hyperbolic sine. Journal of Inequalities and Applications 2008, 2008:-5.Google Scholar
  27. Ogilvy CS, Oppenheim A, Ivanoff VF, Ford, LF Jr., Fulkerson DR, Narayanan, VK Jr.: Elementary problems and solutions: problems for solution: E1275-E1280. The American Mathematical Monthly 1957, 64(7):504–505. 10.2307/2308467MathSciNetView ArticleGoogle Scholar
  28. Oppenheim A, Carver WB: Elementary problems and solutions: solutions: E1277. The American Mathematical Monthly 1958, 65(3):206–209. 10.2307/2310072MathSciNetView ArticleGoogle Scholar
  29. Zhu L: A solution of a problem of oppeheim. Mathematical Inequalities & Applications 2007, 10(1):57–61.MATHMathSciNetView ArticleGoogle Scholar

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© L. Zhu. 2010

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