A Note on Jordan, Adamović-Mitrinović, and Cusa Inequalities

Yang, Zhen-Hang; Chu, Yu-Ming

doi:https://doi.org/10.1155/2014/364076

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 364076 | https://doi.org/10.1155/2014/364076

A Note on Jordan, Adamović-Mitrinović, and Cusa Inequalities

Zhen-Hang Yang¹and Yu-Ming Chu¹

Academic Editor: Soon-Yeong Chung

Received06 Nov 2013

Accepted17 Feb 2014

Published31 Mar 2014

Abstract

We improve the Jordan, Adamović-Mitrinović, and Cusa inequalities. As applications, several new Shafer-Fink type inequalities for inverse sine function and bivariate means inequalities are established, and a new estimate for sine integral is given.

1. Introduction

The classical Jordan inequality [1] is given by for .

Some new developments on refinements, generalizations, and applications for the Jordan inequality can be found in [2] and the references therein.

In the recent past, the following two-side inequality has attracted the attention of many researchers (see, e.g., [2–14]). The left inequality in (2) was obtained by Mitrinović (see [1, p. 238]), while the right one is due to Huygens (see, e.g., [15]) and it is called Cusa inequality [3, 5, 6, 8, 14].

In [16], the following open problem was proposed: for each , there are greatest value and least value such that the double inequality holds for all . This was answered by Carver in [17]. In [1, p. 238, 3.4.15], it was listed that for and . Wu [18] proved that the double inequality holds for , , and . In particular, he obtained that for , The first inequality in (6) is equivalent to the Huygen inequality: Jiang [19] showed that for , Li and He [20] gave an improvement of (6) as follows:

The main purpose of this paper is to give sharp bounds for in terms of the functions and , where

The rest of this paper is organized as follows. Several lemmas are given in Section 2. Main results and their proofs are given in Section 3, in which Theorem 7 unifies and generalizes Jordan and Cusa inequalities; Theorem 13 shows that Adamović-Mitrinović and Cusa inequalities (2) can be interpolated by for suitable ; Theorem 18 gives a hyperbolic version of Theorem 7. In Section 4, some new Shafer-Fink type inequalities for inverse sine function and several inequalities for bivariate means are presented, and a simpler but more accurate estimate for sine integral is provided.

2. Lemmas

Lemma 1. Let and be defined by (10) and (11), respectively. Then and are, respectively, increasing and decreasing with respect to on with the limits

Proof. From (10) and (11) we have If , then we clearly see that . If , then , and then .
Simple computations give (12).

Lemma 2. Let , be defined on by respectively. Then , .

Proof. It is not difficult to see that , for . If , then and then . It remains to prove that for . Differentiation leads to Hence, , which implies that is decreasing in on , and therefore, This completes the proof.

Lemma 3. Let be defined on by Then(i) for all if and only if , where is the unique solution of the equation ;(ii) if and only if ;(iii)for every , there exists a unique such that for and for .

Proof. In order to prove the desired results, we need to rewrite as We clearly see that
We claim that there exists unique such that for and for . Indeed, we have which implies that is increasing on and decreasing on . Since there exists unique such that and for and for . An easy calculation reveals that .
(i) Now we prove the necessary and sufficient condition for for all . Since , we assume that . Denote the minimum point of by . Then . And then, due to , for all if and only if at least one of the following cases occur.
Case 1. Consider that , . It is derived that .
Case 2. Consider that , . It implies that .
Case 3. Consider that , . It yields .
To sum up, for all if and only if .
(ii) It is clear that if and only if and . Solving the inequalities for leads to .
(iii) In the case of , we clearly see that , , and . This implies that there exists a unique such that for and for .
This completes the proof.

Now let us consider the sign of function defined on by where and are defined by (14) and (15), respectively. We have the following.

Lemma 4. Let be defined on by (24). Then(i) for all if and only if ;(ii) for all if and only if , where (iii)in the case of , there exists a unique such that for and for .

Proof. We first give two limit relations as follows: where In fact, if , then making use of power series we get which implies the first relation. Direct computations yield the second one.
Differentiating with respect to leads to where and are defined by (14) and (15), respectively, and here is defined by (19) and .
(i) We now prove that for all if and only if . The necessity easily follows from the inequalities and if and together with the relation (26).
Next we prove the sufficiency. If , then by Lemma 3 , and then . This indicates that is decreasing in on , and therefore, we get . If , then and , which yields . This also yields that is decreasing in on , and so .
(ii) Similarly, we can prove that for all if and only if . If for all , then we have and , which together with (26) and lead to .
In order to prove the sufficiency, we distinguish two cases.
In the case of , by Lemma 3 we have , which implies that is increasing in on , and so, .
In the case of , from Lemma 3 there is a unique such that for and for . This in conjunction with (30) and (29) shows that is decreasing in on and increasing on , and consequently, we have which proves the sufficiency.
(iii) In the case when , we have seen that is decreasing in on and increasing on and for , but Thus, there is a unique such that for and for .
The whole proof is complete.

We next observe the function defined on by Differentiation yields that where , , and are defined by (14), (15), and (24), respectively. From Lemmas 2 and 4 the following assertion is immediate.

Lemma 5. Let be the function defined on by (33). Then(i) is decreasing in on if and only if ;(ii) is increasing in on if and only if , where is given by (25);(iii)in the case when , there is a unique such that is increasing in on and decreasing on .

Lastly, for later use, we also give the following.

Lemma 6. Let be defined on by (10). Then if and only if , and if and only if .

Proof. For , we define Then holds for all if and only if .
In fact, if and only if at least one case of the following occurs.
Case 1. Consider that , . It is impossible.
Case 2. Consider that , . It indicates .
Case 3. Consider that , . It is impossible.
In the same way, we can prove that holds for all if and only if .
We now prove that if and only if . Factoring yields If , then , and then, if and only if , which is equivalent to . If , then , and then, if and only if , which is equivalent to . Consequently, if and only if .
Next we show that if and only if . In fact, if , then if and only if , which yields . It is clearly a contradiction. If , then the statement in question if and only if , which leads to . Thus the proof is complete.

3. Main Results

Theorem 7. Let . Then for , holds if and only if . Moreover, we have for , where is the best possible. And the lower and upper bounds in (38) are decreasing and increasing in on , respectively.

Proof. Clearly, the desired result is equivalent to if and only if , where is defined by (33). To this end, we give two limit relations. The first one follows by expanding in power series for . We have which yields The second one is derived by a simple computation; that is, Now we prove that for all if and only if .
The necessity easily follows by solving the simultaneous inequalities: which implies .
The sufficiency is due to Lemma 5. In fact, If , then by Lemma 5 we see that is decreasing in on . Hence, .
Utilizing the monotonicity of in on gives (38). And from Lemma 1 it is seen that the lower and upper bounds in (38) are decreasing and increasing in on , respectively.
Thus the proof is finished.

By Theorem 7 and Lemma 1, we have the following interesting chain of inequalities.

Corollary 8. For , one has

Remark 9. It is clear that our results unify and refine Jordan and Cusa's inequalities and show that the first one in (9) is sharp. Also, Theorem 7 contains other known results, for example, taking in (38) we get which contain (6). After a simple transformation, (44) can be written as where the second inequality in (45) is due to Neuman and Sándor [6, ].

Theorem 10. Let . Then for holds if and only if , where .
Moreover, for , one has where , is the best possible. And , are decreasing and increasing in on , respectively.
For one has where is the best possible and is the unique root of the equation on .

Proof. Since the inequality (46) is equivalent to , it suffices to prove that holds for if and only if .
Similarly, solving the simultaneous inequalities and with yields , which proves the necessity.
Conversely, the condition is also sufficient for to be valid. For this end, we divide the proof into two cases.
Case 1. Consider that . By Lemma 5 it is seen that is increasing in on , which indicates that .
Case 2. Consider that . By Lemma 5 we see that there is a unique such that is increasing in on and decreasing on . It is acquired that that is, which proves the sufficiency.
In the first case, application of the monotonicity of in on leads to (47), and . In the second case, (51) also yields (47), and Thus we complete the proof.

Remark 11. Taking in (46), we get the first inequality in (9).

Letting and solving (49) by mathematical computation software, we find that and . Letting be defined by (25) yields . By Theorem 10 we get the following.

Corollary 12. For , one has where and are the best possible constants.

Letting in Lemma 6 and using Theorems 7 and 10, we obtain a chain of inequalities that interpolates Adamović-Mitrinović and Cusa's inequalities (2) by .

Theorem 13. For , the inequalities hold if and only if , , , and , where .

Using the monotonicity of in on given by parts one and two of Lemma 5, we see that hold for . And then we have It is clear that the right-hand in (56) is increasing in on , but the monotonicity of left-hand is to be checked. We define where . Logarithmic differentiation leads to where Since we have . Consequently, for .

The result can be stated as a theorem.

Theorem 14. Let . Then for the inequalities hold if and only if , where is the best constant. And the right-hand and left-hand in (61) are increasing and decreasing in , respectively. Inequality (61) is reversed if and only if , where is defined by (25).

Putting in Theorem 14 we have the following.

Corollary 15. For the following inequalities hold:

Further, let be defined on by where is defined by (10). We can show that the monotonicity of in for certain fixed . Differentiation again yields It is easy to verify that Consequently, we have It is reversed for . From these we can obtain the following.

Theorem 16. For the following inequalities hold:

Additionally, Lemma 4 implies an optimal two-side inequality.

Theorem 17. Let and let and be defined by (14) and (15), respectively. Then for the two-side inequality holds if and only if and , where . And, for , the function is decreasing on .

Proof. Since for and by Lemma 2 and defined by (24) can be written as it follows from Lemma 4 that (71) holds if and only if and . It remains to check the monotonicity of in . Differentiation yields where . If , then the numerator of the fraction in right-hand above is clearly positive. Consider that . If , then and , which yields that the numerator is nonnegative.
This proves the assertion.

Similarly, we can obtain a hyperbolic version of Theorems 7 and 10

Theorem 18. Let . Then for holds if and only if . It is reversed if and only if .

Proof. Let be the function defined on by Then the inequalities (74) are equivalent to . Expanding in power series yields which implies On the other hand, we have
Now we prove desired results.
(i) We first prove that holds if and only if .
If for all , then we have Solving the inequalities yields .
We prove the condition is sufficient for to hold for . Differentiation gives where .
Due to , we see that , which yields
Then ; that is, is increasing in on . It is obtained that , which proves the sufficiency.
(ii) Next we prove that the reverse inequality of (74) holds if and only if . The necessity follows from and the assumption . We get .
Now we prove when . We have where , then .
Thus the proof of Theorem 18 is complete.

Denote It is easy to verify that for . By Lemma 1, we see that is decreasing in on . Thus, as a consequence of Theorem 14, we have the following.

Corollary 19. One has

Furthermore, note that and by Lemma 6 we have the following.

Corollary 20. One has where .

4. Applications

In this section, we give some applications of our results.

4.1. Shafer-Fink Type Inequalities

In [1, p. 247, ], it was listed that the inequality holds for , which is due to Shafer [21]. Fink [22] proved that the double inequality is true for . There has been some improvements and generalizations of Shafer-Fink inequality (see [23]). Letting in Theorems 7, 10, 13, 14, 16 and 17 we can obtain corresponding Shafer-Fink type inequalities, which clearly contain many known results. For example, Theorems 7 and 10 can be changed into the following.

Proposition 21. For , the two-side inequality holds if and only if , where is the best possible. And, the lower and upper bounds in (89) are decreasing and increasing in on , respectively.
Inequality (89) is reversed if , where is defined by (25).

Letting , then . Theorem 14 can be restated as follows.

Proposition 22. For , the two-side inequality holds if and only if , where is the best constant. And, the lower and upper bounds in (90) are decreasing and increasing in on , respectively.
Inequality (90) is reversed if , where is defined by (25).

As another example, Theorem 16 can be rewritten as follows.

Proposition 23. For , all the following chains of inequalities hold:

Remark 24. Inequalities (92) are due to Zhu [23].

4.2. Inequalities for Certain Means

For , with , the first and second Seiffert means [24, 25]; Nueman-Sándor means [26] are defined by respectively. More new means can be found in [27]. We also denote the logarithmic mean, arithmetic mean, geometric mean, and quadratic mean of and by , , , and . There has been some inequalities for these means; we quote [7, 26–36]. Now we establish some new ones involving these means.

Let , . Then , ; , . And then Theorems 7, 10, 13, 14, 16 and 17 can be stated as equivalent ones involving means , , , and , . For example, from Theorems 7 and 6 we have the following.

Proposition 25. For with , both the two-side inequalities hold if and only if and , where .

Making changes of variables , yield , ; , , respectively. And then, Theorem 18 can be equivalently written as follows.

Proposition 26. For , with , both the inequalities hold if and only if . They are reversed if and only if .

4.3. The Estimate for the Sine Integral

For the estimations for the sine integral defined by there has been some results (see [37–39]). By our results we can obtain many estimates for . Here we give a simpler but more accurate one.

Proposition 27. For , we have

Proof. By (63) we see that the inequalities hold for . Integrating both sides over and simple calculation yield (98).

Remark 28. By (98) we have

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 11171307 and 61374086 and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.

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Copyright © 2014 Zhen-Hang Yang and Yu-Ming Chu. 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.

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