Sharp Inequalities for Trigonometric Functions

Yang, Zhen-Hang; Jiang, Yun-Liang; Song, Ying-Qing; Chu, Yu-Ming

doi:https://doi.org/10.1155/2014/601839

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 601839 | https://doi.org/10.1155/2014/601839

Sharp Inequalities for Trigonometric Functions

Zhen-Hang Yang,¹Yun-Liang Jiang,²Ying-Qing Song,¹and Yu-Ming Chu¹

Academic Editor: Chun-Gang Zhu

Received04 Mar 2014

Revised21 May 2014

Accepted30 May 2014

Published07 Jul 2014

Abstract

We establish several sharp inequalities for trigonometric functions and present their corresponding inequalities for bivariate means.

1. Introduction

A bivariate real value function is said to be a mean if for all . is said to be homogeneous if for any .

Remark 1 (see [1]). Let be a homogeneous bivariate mean of two positive real numbers and . Then where .

By this remark, almost all of the inequalities for homogeneous symmetric bivariate means can be transformed equivalently into the corresponding inequalities for hyperbolic functions and vice versa. More specifically, let , , and be the logarithmic, identric, and th power means of two distinct positive real numbers and given by respectively. Then, for , we have where . By Remark 1, we can derive some inequalities for hyperbolic functions from certain known inequalities for bivariate means mentioned previously. For example, (see [2, 3]); consider (see [4, 5]); consider that (see [1]) holds for if and only if and ; consider (see [6]); consider (see [7], ( 3.9), and ( 3.10)); if , then the double inequality (see [8]) holds if and only if and ; if , then inequality (11) holds if and only if and ; consider that (see [9]) holds if and only if and .

The main purpose of this paper is to find the sharp bounds for the functions , which include the corresponding trigonometric version of the inequalities listed above. As applications, their corresponding inequalities for bivariate means are presented.

2. Lemmas

Lemma 2 (see [10, Theorem 1.25], [11, Remark 1]). For , let be continuous on and differentiable on ; let on . If is increasing (or decreasing) on , then so are If is one-to-one, then the monotonicity in the conclusion is strict.

Lemma 3 (see [12]). Let and be real numbers and let the power series and be convergent for . If , for , and is increasing (decreasing), for , then the function is also (strictly) increasing (decreasing) on .

Lemma 4 (see [13, pages 227–229]). One has where is the Bernoulli number.

Lemma 5. For every , , the function defined by is increasing if and decreasing if . Consequently, for , one has It is reversed if .

Proof. For , we define and , where . Note that , and can be written as Differentiation and using (14) and (15) yield where Clearly, if the monotonicity of is proved, then by Lemma 3 we can get the monotonicity of , and then the monotonicity of the function easily follows from Lemma 2. For this purpose, since , , for , we only need to show that is decreasing if and increasing if . Indeed, an elementary computation yields It is easy to obtain that, for , which proves the monotonicity of .
Making use of the monotonicity of and the facts that we get inequality (19) and its reverse immediately.

Lemma 6. For every , , the function defined by is increasing if and decreasing if . Consequently, for , one has It is reversed if .

Proof. We define and , where . Note that , and can be written as Differentiating and using (14) and (15) yield where Similarly, we only need to show that is decreasing if and increasing if . In fact, simple computation leads to It is easy to obtain that, for , which proves the monotonicity of .
Making use of the monotonicity of and the facts that we get inequality (27) and its reverse immediately.

Lemma 7 (see [14, 15]). For and , let , , , and be defined by Then, , , and are decreasing with respect to , while is increasing with respect to on .

Proof. It was proved in [14, 15] that the functions and are decreasing with respect to . Now, we prove that has the same property. Logarithmic differentiation gives that, for , Clearly, for and , which yields , and so . This gives and .
Similarly, we get which implies that is decreasing with respect to on . Therefore, which proves the desired result.

3. Main Results

3.1. The First Sharp Bounds for

In this subsection, we present the sharp bounds for in terms of , which give the trigonometric versions of inequalities (6) and (7).

Theorem 8. For , the two-side inequality holds with the best possible constants and , where is the unique root of the equation on . Moreover, one has where the exponents , and coefficients , in (43) are the best possible constants and so is in (44).

Proof. (i) We first prove that the left inequality in (41) for and is the best possible constant. Letting in (19), then we get the first inequality in (41) and the second inequality in (43). If there exists such that for , then Using power series expansion gives Therefore, which derives a contradiction. Hence, is the best possible constant.
(ii) From Lemma 7, we clearly see that the function is decreasing on . Note that Therefore, (42) has a unique root . Numerical calculation gives . Letting in Lemma 5 yields The above inequalities can be rewritten as where the equality is due to the fact that is the unique root of (42). Therefore, we get the right inequality in (41) and the first inequality in (44). We clearly see that is the best possible constant.
(iii) The third inequality in (43) easily follows from which holds due to and . From we clearly see that the coefficients and are the best possible constants.
This completes the proof.

Recently, Yang [16] proved that the inequalities hold for if and only if and , where . Making use of Theorem 8 and Lemma 7, we have the following.

Corollary 9. For , the chain of inequalities hold with the best possible constants , , , and .

3.2. The Second Sharp Bounds for

In this subsection, we give the sharp bounds for in terms of , which give the trigonometric versions of inequalities (8).

Theorem 10. For , the two-side inequality holds with the best possible constants and , where is the unique solution of the equation on . Moreover, the inequalities hold for , where the exponents and the coefficients are the best possible constants. Also, the first member in (57) is decreasing with respect to on , while the third and fourth members are increasing with respect to on . The reverse inequality of (57) holds if .

Proof. For and , we define To prove the desired results, we need two assertions. The first one is which follows by expanding in power series The second one states that the equation , that is, (56), has a unique solution such that for and for . Indeed, Lemma 7 implies that is increasing on , which together with the facts that indicates the second assertion. By using mathematical software, we find .
(i) Now, we prove that the first inequality in (55) holds with the best constant . Letting in Lemma 5 yields the first inequality in (55). Due to the decreasing property of on given by Lemma 7, we assume that there is a with such that the left inequality in (55) holds for ; then we have , which together with the relation (61) leads to . It is clearly impossible. Hence, is the best constant.
(ii) We next show that the second inequality in (55) holds with the best constant . Let us introduce an auxiliary function defined on by Expanding in power series gives where Therefore, we have Differentiation again yields We claim that for . It suffices to show that for . In fact, , and satisfies the recursive relation A direct check leads to due to and satisfies the recursive relation Hence, is decreasing for , and so which yields . From the recursive relation (69), we get for , which proves that for . Note that We also assert that . If not, that is, , then there must be for , which yields and due to being the solution of the equation . This is obviously a contradiction. It follows that there is a such that for and for , which also implies that is decreasing on and increasing on . Therefore, that is, for .
It remains to prove that is the best possible constant. If there is a with such that the right inequality in (55) holds for , then, by the second assertion proved previously, we have , which yields a contradiction.
(iii) The first and second inequalities in (57) and their reverse ones are clearly the direct consequences of Lemma 5. It remains to prove the third one. We have to determine the sign of defined by for and . Arranging leads to As shown previously, for and for , which together with and gives the desired result.
Lemma 7 reveals that the monotonicity of the first, second, and third members in (57) with respect to on due to Finally, we show that is the best possible constant. It easily follows that
Thus, we complete the proof.

Remark 11. Letting and in Theorem 10 and then taking squares, we deduce that the two-side inequality holds for , where .
From the proof of Theorem 10, we clearly see that the constant in (79) is the best possible constant, but is not.

In [15, Theorems 1, 2, and 3], Yang proved that the chain of inequalities holds for with the best constants and . The monotonicity of the function on given in Lemma 7 and Remark 11 lead to the following.

Corollary 12. For , the chain of inequalities holds with the best possible constants , , , and , and .

Using certain known inequalities and the corollary above, we can obtain the following novel inequalities chain for trigonometric functions.

Corollary 13. For , one has

Proof. The first, second, and third inequalities in (82) are due to Neuman [17, Theorem 1].
The fourth one in (82) is equivalent to which holds due to for .
The eighth one is derived from Neuman and Sándor [18, ( 2.5)].
The ninth one easily follows from The tenth, eleventh, and twelfth ones can be obtained by [19, ( 3.9)].
Except the last one, other ones are obviously deduced from Corollary 12.
The last one is equivalent to which follows from the inequality connecting the fourth and sixth members in(82) proved previously.
Thus, the proof is complete.

Remark 14. Sándor [20, page 81, Lemma 2.2] proved that the inequality holds for . Clearly, the sixth and seventh inequalities in (82), that is, for , are a refinement of Sándor's inequality.

Remark 15. Using the decreasing property of the function defined by (83) proved in Corollary 13, we also get for , which can be rewritten as This in conjunction with (43) gives From we conclude that and are also the best possible constants.
Further, we conjecture that hold for , where all exponents are optimal.

Taking , , and in (57), we get the following.

Corollary 16. For , we have where , and , are the best possible constants.

Remark 17. The inequalities connecting the first, fourth, and seventh members in (93) state that, for , which can be written as or It is easy to check that this double inequality is stronger than the new Redheffer-type one for proved by Zhu and Sun [21, Theorem 3]; that is, for ,

Remark 18. Making use of the double inequalities for proved in [22] and [15, Corollary 3], respectively and taking into account (93) and (94), we easily obtain

3.3. The Sharp Bounds for

In this subsection, we establish sharp inequalities between and and prove the trigonometric version of inequalities (9) and (10). Employing Lemmas 6 and 7, we have the following.

Theorem 19. For , the two-side inequality holds with the best possible constants and , where is the unique root of the equation on . Moreover, the inequalities hold for , where the exponents and the coefficients are the best possible constants. Also, the first member in (104) is decreasing with respect to on , while the third and fourth members are increasing with respect to on . The reverse of (104) holds if .

Proof. For and , we define To prove the desired results, we need two assertions. The first is the limit relation which follows by expanding in power series The second one states that the equation , that is, (103), has a unique solution such that for and for . In fact, Lemma 7 implies that is increasing on , which in conjunction with the facts that indicates the second one. By using mathematical software, we find .
(i) Now we show that the first inequality in (102) holds for with the best constants . In fact, the first inequality in (102) follows by Lemma 6. On the other hand, due to the decreasing property of with respect to on , if there is a smaller with such that the first inequality in (102) holds for , then there must be , which by the relation (108) gives . This yields a contradiction. Consequently, the constants is optimal.
(ii) We next prove that the second inequality in (102) holds for , where is the best possible constant. We introduce an auxiliary function defined on by Expanding in power series leads to where Therefore, we have Differentiation again yields We assert that for . It suffices to show that for . In fact, , and satisfies the recursive relation A direct check gives , due to and satisfies the recursive relation Hence, is decreasing for , and so which yields . From the recursive relation (116), we get for , which proves that for . Therefore, we get Next, we divide the proof into two cases.
Case 1 . In this case, we clearly see that for and for . Hence, , and so for , which reveals that and for , where due to being the unique root of (103). This is impossible.
Case 2 . In this case, we see that there is a such that for and for . This indicates that is decreasing on and increasing on . Thus, we have for .
If , then for . Similar to Case 1, this also yields a contradiction.
If , then there is a such that for and for , which together with (111) shows that is decreasing on and increasing on . Therefore, that is, for .
On the other hand, if there is a with such that the second inequality in (55) holds for , then by the second assertion proved previously, we have , which leads to a contradiction. This proves that the constant is the best possible constant.
(iii) The first and second inequalities in (57) and their reverse ones are clearly the direct consequences of Lemma 6. It remains to prove the third one. We have to determine the sign of defined by for and . Simplifying leads to As shown previously, for and for , which in combination with and gives the desired result.
Lemma 7 reveals the monotonicity of the first, second, and third members in (104) with respect to on due to Finally, we prove that is the best possible constant. It can be deduced from
Thus, the proof is complete.

We note that (102) can be written as Making use of the monotonicity of the function on given in Lemma 7 together with Corollary 12 and Theorem 19, we obtain the following.

Corollary 20. For , the chain of inequalities holds, where , , , , , , and are the best possible constants, and .

Remark 21. From the above corollary, we clearly see that for . The relation connecting the first, third, and fourth members in (128) can be written as

Taking , in Theorem 19, we have the following.

Corollary 22. For , the inequalities hold, where the exponents and and the coefficients and are the best possible constants.

Theorem 23. For , we have where , , , , and are the best possible constants.

Proof. (i) We first prove (132). For this purpose, let us define Differentiating gives where Using double angle formula and Lemma 4, we have Hence, for , and so which implies the desired inequalities.
(ii) Now, we prove (133). Differentiation leads to where the inequality holds for due to (88). Therefore, which deduces (133).
(iii) Similarly, we have which gives

Using inequalities (129) and (133), we get immediately the trigonometric version of (9).

Corollary 24. For , we have

3.4. The Third Sharp Bounds for

The trigonometric versions of (11) and (12) are contained in the following theorem.

Theorem 25. Let . Then the following statements are true:(i)if , then the two-side inequality holds if and only if and ;(ii)if , then the double inequality (145) holds if and only if and ;(iii)if , then the double inequality (145) holds if and only if and ;(iv)the double inequality holds if and only if and , where is the weighted power mean of order of and defined by

Proof. For and , we define Since , can be written as Differentiation gives where Clearly, if we prove that for and for with , then, by Lemma 2, we know that is increasing if and decreasing if with , and which yield the first, second, and third results in this theorem.
Now, we show that if and if with . Simple computations lead to for . Using (15)–(17), we have By Lemma 3, in order to prove the monotonicity of , it suffices to get the monotonicity of . Note that Differentiating , we get for . The function is decreasing on , and we conclude that Thus, if and if with .
Finally, we prove the fourth result. The first part implies that the right-hand side inequality in (146) holds if . While the necessity can be obtained from the following limit relation: in fact, power series expansion leads to
Now, we prove that the left-hand side inequality holds if and only if . The necessity follows easily from Next, we deal with the sufficiency. We divide the proof into two cases.
Case 1 . The sufficiency follows immediately from the second and third results proved previously.
Case 2 . It was proved previously that the function is decreasing on , and so the function is increasing on the same interval. The monotonicity together with leads to the conclusion that there exists unique such that for and for ; then, from (151), we know that is decreasing on and increasing on . It follows from Lemma 2 that is decreasing on , and so we have which can be rewritten as On the other hand, Lemma 2 also implies that is increasing on . Therefore, which implies that Clearly, if we can prove that the right-hand side in (167) is also greater than the right-hand side in (164), then the proof is completed. Since satisfies (164), for , we have where the last inequality holds due to and .
Thus, the proof is finished.

4. Some Corresponding Inequalities for Means

The Schwab-Borchardt mean of two numbers and is defined by (see [23, Theorem 8.4], [24, ], and [25, ]). It is clear that is not symmetric in its variables and is a homogeneous function of degree 1 in and . More properties of this mean can be found in [25–27]. Very recently, Yang [19, Definitions 3.2, 4.2, and 5.2] defined three families of two-parameter trigonometric means. For convenience, we recall the definition of two-parameter sine mean as follows.

Definition 26. Let and such that , and let be defined by Then defined by is called a two-parameter sine mean of and .

In particular, for , are means of and . Similarly, according to the definition of two-parameter cosine mean (see [19, Definition 4.2]), is also a mean of and , where is defined by (34).

Further, we have the following.

Proposition 27. For and , the function is also a mean of and , where is defined by (35).

Proof. It suffices to prove that the double inequality holds for , which is equivalent to where .
Using the decreasing property proved in Lemma 7, we see that which proves the assertion.

If we replace by and then multiply or for suitable in each sides of the inequalities in previous section, then we can get the corresponding inequalities for bivariate means. For example, Theorems 10, 19, and 25 can be rewritten as follows.

Theorem . For , the two-side inequalityholds with the best possible constants and , where is the unique solution of (56) on .

Theorem . For , the two-side inequality holds with the best constants and , where is the unique root of (103) on .

Theorem . Let .Then the following statements are true:(i) if , then the two-side inequalityholds if and only if and ;(ii) if , then the double inequality (180) holds if and only if and ;(iii) if , then the double inequality (180) holds if and only if and ;(iv) the double inequality holds if and only if and , where the left hand side in (181) is defined as if .

Similar to , these bivariate means mentioned previously are not symmetric in their variables and are homogeneous of degree 1 in and . But they can generate more symmetric means by making certain substitutions; for example, Neuman and Sándor [25, ] proved that , , where , , , , and denote the quadratic, arithmetic, geometric, first, and second Seiffert means [28, 29] of and given by respectively. In same way, we have which is a Sándor mean introduced in [20, page 82], [30]. Also, we get which is also a new mean, and it satisfies the double inequality .

There are many inequalities involving means , , , , and ; we quote [15, 20, 25, 27, 31–44]. Inequalities for Sándor's mean can be found in [20, pages 86–93] and [19, Section 6].

We now deduce some inequalities involving these means from the inequalities for trigonometric functions established in Section 3.

Step 1. Put , where .

Step 2. Put , where are means of positive numbers and , and for all .

Let and . Then the following variable substitutions follows from Steps 1 and 2.

(i) Substitution 1: . Then

(ii) Substitution 2: . Then For simplicity in expressions, we only select the functions involving , , and in a chain of inequalities given in Section 3.

The following follows from (88).

Proposition 28. For , the inequalities hold. Moreover, replacing by , we have replacing by , we get

Remark 29. The second inequalities in (188) and (189) are due to Sándor [31, 33], while the one connecting and first appeared in [20, page 82, ].

From inequalities (90), we have the following.

Proposition 30. For , the double inequality is valid, where and are the best possible constants. Moreover, replacing by and , we get

Remark 31. The left-hand side inequalities in (190), (191), and (192) can be found in [19, Example 6.1]. But the left-hand side inequality in (191) is weaker than proved by Sándor [20, page 89, ].

Inequalities (100) can be written as the corresponding inequalities for certain bivariate means as follows.

Proposition 32. For , the inequalities hold true with the best possible exponents and coefficients. Moreover, replacing by and , we have

Remark 33. The fourth inequality in (195) was first proved by Sándor in [31].

From (130) in Corollary 22, we clearly see the following.

Proposition 34. For , the inequalities hold, where the exponents and and the coefficients and are the best possible constants. Moreover, replacing by and , we have

Inequalities (134) lead to the following.

Proposition 35. For , the sharp inequalities hold true. Moreover, replacing by , , we have

Inequalities (144) lead to the following conclusion.

Proposition 36. For , the inequalities are valid, where is the best possible constant. In particular, replacing by and , we get

Remark 37. The first inequality in (202) was established by Sándor in [20, page 87, ].

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 11371125 and 61374086, the Natural Science Foundation of Hunan Province under Grant 14JJ2127, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.

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Copyright © 2014 Zhen-Hang Yang et al. 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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