Research Article | Open Access

# A Sharp Double Inequality for Trigonometric Functions and Its Applications

**Academic Editor:**Josip E. Pečarić

#### Abstract

We present the best possible parameters and such that the double inequality holds for any . As applications, some new analytic inequalities are established.

#### 1. Introduction

It is well known that the double inequality holds for any . The first inequality in (1) was found by Mitrinović (see [1]), while the second inequality in (1) is due to Huygens (see [2]) and it is called Cusa inequality. Recently, the improvements, refinements, and generalizations for inequality (1) have attracted the attention of many mathematicians [3–8].

Qi et al. [9] proved that the inequality holds for any . It is easy to verify that and cannot be compared on the interval .

Neuman and Sándor [6] gave an improvement for the first inequality in (1) as follows:

Inequality (3) was also proved by Lv et al. in [10]. In [11, 12], Neuman proved that the inequalities hold for any .

For the second inequality in (1), Klén et al. [13] established for .

Inequality (5) was improved by Yang [14]. In [15], Yang further proved for .

Yang [16] proved that the inequalities hold for .

Zhu [8] and Yang [17] proved that and are the best possible constants such that the double inequality holds for all .

More results involving inequality (1) can be found in the literature [18–22].

Let , , and . Then is defined by

It is well known that is strictly increasing with respect to for fixed and (see [23]). If , then it is easy to check that

It follows from (2) and (3) together with (6) that for .

The main purpose of this paper is to present the best possible parameters and such that the double inequality holds for all . As applications, some new analytic inequalities are found. All numerical computations are carried out using MATHEMATICA software.

#### 2. Lemmas

In order to prove our main results we need several lemmas, which we present in this section.

Lemma 1. *Let and the function be defined on by
**Then the following statements are true:*(i)* for all if and only if ;*(ii)* for all if and only if , where is the unique solution of equation
*(iii)*if , then there exists such that for and for .*

*Proof. *It follows from (13) and (14) that
for .

Inequalities (15) lead to the conclusion that the function is strictly decreasing with respect to for fixed and is the unique solution of (14).

(i) If and , then from the monotonicity of the function we clearly see that

If for all , then (13) leads to

(ii) If and , then the monotonicity of the function leads to the conclusion that .

If and , then (13) and the monotonicity of the function lead to

Inequality (19) implies that the function is concave with respect to on the interval . Therefore, follows from (18) and the concavity of .

If for all , then follows easily from the monotonicity of the function and together with the fact that .

(iii) If and , then from (13) and (19) together with the monotonicity of the function we get
and is strictly decreasing on .

It follows from (21) and (22) together with the monotonicity of that there exists such that is strictly increasing on and strictly decreasing on . Therefore, Lemma 1 follows from (20) and the piecewise monotonicity of .

Let and the function be defined on by

Then elaborated computations lead to
where is defined by (13).

From Lemma 1 and (24) we get the following Lemma 2 immediately.

Lemma 2. *Let and be defined on by (23). Then*(i)* is strictly decreasing on if and only if ;*(ii)* is strictly increasing on if and only if , where is the unique solution of (14);*(iii)*if , then there exists such that is strictly increasing on and strictly decreasing on .*

Lemma 3. *Let and be defined on by (23). Then*(i)* for all if and only if ;*(ii)* for all if and only if ;*(iii)*if , then there exists such that for and for .*

*Proof. *(i) If and , then from (23) and Lemma 2 we clearly see that

If for all , then (23) leads to
(ii) If for all , then from (23) we get

Inequality (27) leads to the conclusion that .

If and , then we divide the proof into two cases.*Case 1*. Consider , where is the unique solution of (14). Then from Lemma 2 and (23) we clearly see that*Case 2*. Consider . Then (23) and Lemma 2 lead to
and there exists such that is strictly increasing on and strictly decreasing on . Therefore, for all follows from (29) and the piecewise monotonicity of .

(iii) If , then . It follows from (23) and Lemma 2 that
and there exists such that is strictly increasing on and strictly decreasing on . Therefore, Lemma 3 follows from (30) and the piecewise monotonicity of .

Let and be defined on by

Then elaborated computations give
where is defined by (23).

From Lemma 3 and (33) we get Lemma 4 immediately.

Lemma 4. *Let and be defined on by (31) and (32). Then*(i)* is strictly decreasing on if and only if ;*(ii)* is strictly increasing on if and only if ;*(iii)*if , then there exists such that is strictly increasing on and strictly decreasing on .*

Lemma 5. *Let and be defined on by (31) and (32). Then the following statements are true:*(i)*if for all , then ;*(ii)*if for all , then , where is the unique solution of the equation
**on the interval .*

*Proof. * If for all , then from (31) and (32) we have
We first prove that is the unique solution of (34) on the interval . Let and

Then numerical computations show that

Inequality (38) implies that is strictly decreasing on . Therefore, is the unique solution of (34) on the interval which follows from (37) and the monotonicity of .

If and for all , then (31) leads to

Therefore, follows from (39) and together with the monotonicity of on the interval .

Lemma 6. *Let and , and let be defined by (9). Then the function is strictly decreasing with respect to if .*

*Proof. *Let . Then from (9) we get
Inequality (41) and lead to the conclusion that is strictly decreasing with respect to . Therefore, for , and is strictly decreasing with respect to if .

#### 3. Main Results

Theorem 7. *Let be defined by (9). Then the double inequality
**
holds for all if and only if , and the double inequality
**
holds for all if and only if , where
**, and is strictly decreasing with respect to .*

*Proof. *Let and be defined on by (31) and (32). Then

If , then inequality (42) follows from Lemma 4 and (45).

If inequality (42) holds for all , then for all . It follows from Lemma 5 that .

If , then inequality (43) follows from Lemma 4 and (45).

If inequality (43) holds for all , then for all . It follows from Lemma 5 that , where is the unique solution of (34) on the interval . We claim that ; otherwise , and Lemma 4 leads to the conclusion that there exists such that for .

Note that

It follows from Lemma 6 and (46) that is strictly decreasing with respect to .

From Theorem 7 we get Corollaries 8 and 9 as follows.

Corollary 8. *For all one has
*

Corollary 9. *For all one has
*

Theorem 10. *Let be defined by (9). Then the double inequality
**
holds for all if and only if and , where is the unique solution of (34) on the interval . Moreover, the inequality
**
if and only if
**
where is defined as in Lemma 3 .*

*Proof. *Let and be defined on by (31) and (32). Then Lemma 4 leads to the conclusion that is strictly increasing on and strictly decreasing on . Note that

It follows from the piecewise monotonicity of and (52) that
for all . Therefore, for all follows from the first inequality of (53), while for all follows from the second inequality of (42).

Conversely, if the double inequality (49) holds for all , then we clearly see that the inequalities
hold for all . Therefore, and follows from Lemma 5 and (54). Moreover, numerical computations show that and

Therefore, the second conclusion of Theorem 10 follows from (55) and the second inequality of (53).

It follows from Lemma 3 that we get Theorem 11 immediately.

Theorem 11. *The double inequalities
**
hold for all if and only if and .*

We clearly see that the function is strictly decreasing with respect to for fixed . Let and ; then Theorem 11 leads to the following.

Corollary 12. *The inequalities
**
hold for all .*

#### 4. Applications

In this section, we give some applications for our main results.

Neuman [24] proved that the Huygens type inequalities hold for all . Note that if , and the second inequalities in (59) are reversed if .

From Theorems 7 and 10 together with (59) we get the following.

Theorem 13. *The double inequality
**
holds for all if and only if or , and inequality (60) is reversed if and only if .*

Theorem 14. *The double inequality
**
holds for all if and only if and or , where is the unique solution of (34) on the interval .*

Neuman [24] also proved that the Wilker type inequality holds for all if .

Making use of Theorem 13 and the arithmetic-geometric means inequality we get Corollary 15 as follows.

Corollary 15. *The Wilker type inequality (62) holds for all if or .**In addition, power series expansions show that
**
Therefore, we conjecture that inequality (62) holds for all if and only if or . We leave it to the readers for further discussion.**The Schwab-Borchardt mean [25–27] of two distinct positive real numbers and is defined by
**
where and are the inverse cosine and inverse hyperbolic cosine functions, respectively.**Let , be the arithmetic mean of and , and . Then simple computations lead to
**It follows from Theorems 7, 10, and 11 together with (66) that we have the following.*

Theorem 16. *Let , and be defined as in Theorems 7 and 10, respectively. Then for all , the following statements are true.*(i)*The double inequality
holds if and only if , and inequality (67) is reversed if and only if .*(ii)*The double inequality
holds if and only if and .*(iii)*The double inequality
holds if and only if and .*

Let , , , , , and be the geometric, quadratic, first Seiffert [28], second Seiffert [29], and Yang [15] means of and , respectively. Then it is easy to check that , , and . Therefore, Theorem 16 leads to Corollary 17.

Corollary 17. *Let , and be defined as in Theorems 7 and 10, respectively. Then for all , the following statements are true.**
(i) The double inequalities
**
hold if and only if , and all inequalities in (70) are reversed if and only if .**
(ii) The double inequalities
**
hold if and only if and .**
(iii) The double inequalities
*