Sharp Inequalities for Trigonometric Functions

Abstract

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

(ProQuest: ... denotes non-US-ASCII text omitted.)

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

Academic Editor:Chun-Gang Zhu

1, School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China
2, School of Information Engineering, Huzhou Teachers College, Huzhou 313000, China

Received 4 March 2014; Revised 21 May 2014; Accepted 30 May 2014; 7 July 2014

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.

1. Introduction

A bivariate real value function M : ( 0 , ∞ ) × ( 0 , ∞ ) ... ( 0 , ∞ ) is said to be a mean if [figure omitted; refer to PDF] for all x , y > 0 . M is said to be homogeneous if [figure omitted; refer to PDF] for any λ , x , y > 0 .

Remark 1 (see [1]).

Let M ( x , y ) be a homogeneous bivariate mean of two positive real numbers x and y . Then [figure omitted; refer to PDF] where t = ( 1 / 2 ) ln ... ( x / y ) .

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 L ( x , y ) , I ( x , y ) , and _{A r} ( x , y ) be the logarithmic, identric, and r th power means of two distinct positive real numbers x and y given by [figure omitted; refer to PDF] respectively. Then, for x > y > 0 , we have [figure omitted; refer to PDF] where t = ( 1 / 2 ) ln ... ( x / y ) > 0 . By Remark 1, we can derive some inequalities for hyperbolic functions from certain known inequalities for bivariate means mentioned previously. For example, [figure omitted; refer to PDF] (see [2, 3]); consider [figure omitted; refer to PDF] (see [4, 5]); consider that [figure omitted; refer to PDF] (see [1]) holds for t > 0 if and only if p ...5; 2 / 3 and 0 < q ...4; _{q 0} = 10 / 5 ; consider [figure omitted; refer to PDF] (see [6]); consider [figure omitted; refer to PDF] (see [7], ( 3.9), and ( 3.10)); if 0 < p ...4; 6 / 5 , then the double inequality [figure omitted; refer to PDF] (see [8]) holds if and only if _{λ p} ...4; 2 / 3 and _{μ p} ...5; ^{( 2 / e ) p} ; if p ...5; 2 , then inequality (11) holds if and only if _{λ p} ...4; ^{( 2 / e ) p} and _{μ p} ...5; 2 / 3 ; consider that [figure omitted; refer to PDF] (see [9]) holds if and only if p ...4; 6 / 5 and q ...5; ( log ... 3 - log ... 2 ) / ( 1 - log ... 2 ) .

The main purpose of this paper is to find the sharp bounds for the functions ^{e t cot t - 1} ( t ∈ ( 0 , π / 2 ) ) , 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 - ∞ < a < b < ∞ , let f , g : [ a , b ] [arrow right] R be continuous on [ a , b ] and differentiable on ( a , b ) ; let ^{g [variant prime]} ...0; 0 on ( a , b ) . If ^{f [variant prime]} / ^{g [variant prime]} is increasing (or decreasing) on ( a , b ) , then so are [figure omitted; refer to PDF] If ^{f [variant prime]} / ^{g [variant prime]} is one-to-one, then the monotonicity in the conclusion is strict.

Lemma 3 (see [12]).

Let _{a n} and _{b n} ( n = 0,1 , 2 , ... ) be real numbers and let the power series A ( t ) = ^{∑ n = 1 ∞} ... _{a n} ^{t n} and B ( t ) = ^{∑ n = 1 ∞} ... _{b n} ^{t n} be convergent for | t | < R . If _{a n} , _{b n} > 0 , for n = 1,2 , ... , and _{a n} / _{b n} is ( s t r i c t l y ) increasing (decreasing ), for n = 1,2 , ... , then the function A ( t ) / B ( t ) is also (strictly ) increasing (decreasing ) on ( 0 , R ) .

Lemma 4 (see [13, pages 227-229]).

One has [figure omitted; refer to PDF] where _{B n} is the Bernoulli number.

Lemma 5.

For every t ∈ ( 0 , π / 2 ) , p ∈ ( 0,1 ] , the function _{F p} defined by [figure omitted; refer to PDF] is increasing if p ∈ ( 0,1 / 2 ] and decreasing if p ∈ [ 10 / 5,1 ] . Consequently, for p ∈ ( 0,1 / 2 ] , one has [figure omitted; refer to PDF] It is reversed if p ∈ [ 10 / 5,1 ] .

Proof.

For t ∈ ( 0 , π / 2 ) , we define _{f 1} ( t ) = t cot t - 1 and _{f 2} ( t ) = ln ... ( cos ... p t ) , where p ∈ ( 0,1 ] . Note that _{f 1} ( ^{0 +} ) = _{f 2} ( ^{0 +} ) = 0 , and _{F p} ( t ) can be written as [figure omitted; refer to PDF] Differentiation and using (14) and (15) yield [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Clearly, if the monotonicity of _{a n} / _{b n} is proved, then by Lemma 3 we can get the monotonicity of ^{f 1 [variant prime]} / ^{f 2 [variant prime]} , and then the monotonicity of the function _{F p} easily follows from Lemma 2. For this purpose, since _{a n} , _{b n} > 0 , for n ∈ N , we only need to show that _{b n} / _{a n} is decreasing if p ∈ ( 0,1 / 2 ] and increasing if p ∈ [ 10 / 5,1 ] . Indeed, an elementary computation yields [figure omitted; refer to PDF] It is easy to obtain that, for ∈ N , [figure omitted; refer to PDF] which proves the monotonicity of _{a n} / _{b n} .

Making use of the monotonicity of _{F p} and the facts that [figure omitted; refer to PDF] we get inequality (19) and its reverse immediately.

Lemma 6.

For every t ∈ ( 0 , π / 2 ) , p ∈ ( 0,1 ] , the function _{G p} defined by [figure omitted; refer to PDF] is increasing if p ∈ ( 0,1 / 2 ] and decreasing if p ∈ [ 1 / 3 , 1 ] . Consequently, for p ∈ ( 0,1 / 2 ] , one has [figure omitted; refer to PDF] It is reversed if p ∈ [ 1 / 3 , 1 ] .

Proof.

We define _{g 1} ( t ) = ln ... ( sin t / t ) + t cot t - 1 and _{g 2} ( t ) = ln ... ( cos ... p t ) , where p ∈ ( 0,1 ] . Note that _{g 1} ( ^{0 +} ) = _{g 2} ( ^{0 +} ) = 0 , and _{G p} ( t ) can be written as [figure omitted; refer to PDF] Differentiating and using (14) and (15) yield [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Similarly, we only need to show that _{d n} / _{c n} is decreasing if p ∈ ( 0,1 / 2 ] and increasing if p ∈ [ 1 / 3 , 1 ] . In fact, simple computation leads to [figure omitted; refer to PDF] It is easy to obtain that, for n ∈ N , [figure omitted; refer to PDF] which proves the monotonicity of _{c n} / _{d n} .

Making use of the monotonicity of _{F p} and the facts that [figure omitted; refer to PDF] we get inequality (27) and its reverse immediately.

Lemma 7 (see [14, 15]).

For t ∈ [ 0 , π / 2 ] and p ∈ [ 0,1 ) , let _{U p} ( t ) , _{V p} ( t ) , _{W p} ( t ) , and _{R p} ( t ) be defined by [figure omitted; refer to PDF] Then, _{U p} ( t ) , _{V p} ( t ) , and _{W p} ( t ) are decreasing with respect to p ∈ [ 0,1 ) , while _{R p} ( t ) is increasing with respect to p on [ 0,1 ) .

Proof.

It was proved in [14, 15] that the functions _{U p} ( t ) and _{V p} ( t ) are decreasing with respect to p ∈ [ 0,1 ) . Now, we prove that _{W p} ( t ) has the same property. Logarithmic differentiation gives that, for p ∈ ( 0,1 ) , [figure omitted; refer to PDF] Clearly, ^{[varphi] 2 [variant prime]} ( p ) > 0 for t ∈ [ 0 , π / 2 ] and p ∈ ( 0,1 ) , which yields _{[varphi] 2} ( p ) > _{[varphi] 2} ( 0 ) = 0 , and so ^{[varphi] 1 [variant prime]} ( p ) ...4; 0 . This gives _{[varphi] 1} ( p ) < _{[varphi] 1} ( 0 ) = 0 and ∂ ln ... _{W p} ( t ) / ∂ p < 0 .

Similarly, we get [figure omitted; refer to PDF] which implies that ∂ ln ... _{W p} ( t ) / ∂ p is decreasing with respect to t on [ 0 , π / 2 ] . Therefore, [figure omitted; refer to PDF] which proves the desired result.

3. Main Results

3.1. The First Sharp Bounds for ^{e t cot t - 1}

In this subsection, we present the sharp bounds for ^{e t cot t - 1} in terms of ^{( cos ... p t ) 1 / p} , which give the trigonometric versions of inequalities (6) and (7).

Theorem 8.

For t ∈ ( 0 , π / 2 ) , the two-side inequality [figure omitted; refer to PDF] holds with the best possible constants 2 / 3 and _{p 1} = 0.6505 ... , where _{p 1} is the unique root of the equation [figure omitted; refer to PDF] on ( 0,1 ) . Moreover, one has [figure omitted; refer to PDF] where the exponents 3 / 2 , 1 / ln ... 2 and coefficients 1 , 2 2 / e in (43) are the best possible constants and so is _{p 1} [approximate] 0.6505536 in (44).

Proof.

(i) We first prove that the left inequality in (41) for t ∈ ( 0 , π / 2 ) and 2 / 3 is the best possible constant. Letting p = 2 / 3 ∈ [ 10 / 5,1 ] in (19), then we get the first inequality in (41) and the second inequality in (43). If there exists p < 2 / 3 such that ^{e t cot t - 1} > ^{( cos ... p t ) 1 / p} for t ∈ ( 0 , π / 2 ) , then [figure omitted; refer to PDF] Using power series expansion gives [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] which derives a contradiction. Hence, 2 / 3 is the best possible constant.

(ii) From Lemma 7, we clearly see that the function p ... 1 + ( 1 / p ) ln ... ( cos ... ( p π / 2 ) ) is decreasing on ( 0,1 ) . Note that [figure omitted; refer to PDF] Therefore, (42) has a unique root _{p 1} ∈ ( 0,1 ) . Numerical calculation gives _{p 1} [approximate] 0.6505536 . Letting p = _{p 1} ∈ [ 10 / 5,1 ] in Lemma 5 yields [figure omitted; refer to PDF] The above inequalities can be rewritten as [figure omitted; refer to PDF] where the equality is due to the fact that _{p 1} is the unique root of (42). Therefore, we get the right inequality in (41) and the first inequality in (44). We clearly see that _{p 1} is the best possible constant.

(iii) The third inequality in (43) easily follows from [figure omitted; refer to PDF] which holds due to ln ... ( cos ... ( 2 t / 3 ) ) > ln ... ( cos ... ( π / 3 ) ) = - ln ... 2 and 1 / ln ... 2 < 3 / 2 . From [figure omitted; refer to PDF] we clearly see that the coefficients 1 and 2 2 / e are the best possible constants.

This completes the proof.

Recently, Yang [16] proved that the inequalities [figure omitted; refer to PDF] hold for t ∈ ( 0 , π / 2 ) if and only if p ∈ [ _{p 0} , 1 ) and q ∈ ( 0,1 / 3 ] , where _{p 0} [approximate] 0.3473 . Making use of Theorem 8 and Lemma 7, we have the following.

Corollary 9.

For t ∈ ( 0 , π / 2 ) , the chain of inequalities [figure omitted; refer to PDF] hold with the best possible constants 2 / 3 , _{p 1} [approximate] 0.6505 , _{p 0} [approximate] 0.3473 , and 1 / 3 .

3.2. The Second Sharp Bounds for ^{e t c o t t - 1}

In this subsection, we give the sharp bounds for ^{e t cot t - 1} in terms of ^{( cos ... p t ) 2 / ( 3 p 2 )} , which give the trigonometric versions of inequalities (8).

Theorem 10.

For t ∈ ( 0 , π / 2 ) , the two-side inequality [figure omitted; refer to PDF] holds with the best possible constants 2 / 10 and _{p 2} [approximate] 0.6210901 , where _{p 2} is the unique solution of the equation [figure omitted; refer to PDF] on ( 1 / 2,1 ) . Moreover, the inequalities [figure omitted; refer to PDF] hold for p ∈ [ 10 / 5,1 ] , where the exponents [figure omitted; refer to PDF] and the coefficients [figure omitted; refer to PDF] are the best possible constants. Also, the first member in (57) is decreasing with respect to p on ( 0,1 ) , while the third and fourth members are increasing with respect to p on ( 0,1 ) . The reverse inequality of (57) holds if p ∈ ( 0,1 / 2 ] .

Proof.

For t ∈ ( 0 , π / 2 ) and p ∈ ( 0,1 ) , we define [figure omitted; refer to PDF] To prove the desired results, we need two assertions. The first one is [figure omitted; refer to PDF] which follows by expanding in power series [figure omitted; refer to PDF] The second one states that the equation _{H p} ( π / ^{2 -} ) = 0 , that is, (56), has a unique solution _{p 2} [approximate] 0.6210901 such that _{H p} ( π / ^{2 -} ) < 0 for p ∈ ( 0 , _{p 2} ) and _{H p} ( π / ^{2 -} ) > 0 for p ∈ ( _{p 2} , 1 ) . Indeed, Lemma 7 implies that p ... _{H p} ( π / ^{2 -} ) is increasing on ( 0,1 ) , which together with the facts that [figure omitted; refer to PDF] indicates the second assertion. By using mathematical software, we find _{p 2} [approximate] 0.6210901 .

(i) Now, we prove that the first inequality in (55) holds with the best constant 2 / 10 . Letting p = 2 / 10 in Lemma 5 yields the first inequality in (55). Due to the decreasing property of p ... ( cos ... p t ) 2 / ( 3 p 2 ) on ( 0,1 ) given by Lemma 7, we assume that there is a p [variant prime] ∈ ( 0,1 ) with p [variant prime] < 2 / 10 such that the left inequality in (55) holds for t ∈ ( 0 , π / 2 ) ; then we have lim ... t [arrow right] 0 + t - 4 H p [variant prime] ( t ) ...5; 0 , which together with the relation (61) leads to ( p [variant prime] 2 - 2 / 5 ) ...5; 0 . It is clearly impossible. Hence, 2 / 10 is the best constant.

(ii) We next show that the second inequality in (55) holds with the best constant _{p 2} . Let us introduce an auxiliary function _{h p 2} defined on ( 0 , π / 2 ) by [figure omitted; refer to PDF] Expanding in power series gives [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Therefore, we have [figure omitted; refer to PDF] Differentiation again yields [figure omitted; refer to PDF] We claim that ^{h _{p 2} [variant prime]} ( t ) > 0 for t ∈ ( 0 , π / 2 ) . It suffices to show that _{r n} > 0 for n ...5; 3 . In fact, _{r 3} = 63 ( ^{p 2 4} - 1 / 7 ) > 0 , and _{r n} satisfies the recursive relation [figure omitted; refer to PDF] A direct check leads to [figure omitted; refer to PDF] due to ^{r 3 [variant prime][variant prime]} = 55809 and ^{r n [variant prime][variant prime]} satisfies the recursive relation [figure omitted; refer to PDF] Hence, ^{r n [variant prime]} is decreasing for n ...5; 3 , and so [figure omitted; refer to PDF] which yields ^{p 2 2} - ^{r n [variant prime]} > ^{p 2 2} - 28 / 85 > 0 . From the recursive relation (69), we get _{r n} > 0 for n ...5; 3 , which proves that ^{h _{p 2} [variant prime]} ( t ) > 0 for t ∈ ( 0 , π / 2 ) . Note that [figure omitted; refer to PDF] We also assert that _{h p 2} ( π / ^{2 -} ) > 0 . If not, that is, _{h p 2} ( π / ^{2 -} ) ...4; 0 , then there must be ^{H _{p 2} [variant prime]} ( t ) < 0 for t ∈ ( 0 , π / 2 ) , which yields _{H p 2} ( t ) < _{H p 2} ( ^{0 +} ) = 0 and _{H p 2} ( t ) > _{H p 2} ( π / ^{2 -} ) = 0 due to _{p 2} being the solution of the equation _{H p} ( π / ^{2 -} ) = 0 . This is obviously a contradiction. It follows that there is a _{t 1} ∈ ( 0 , π / 2 ) such that _{h p 2} ( t ) < 0 for t ∈ ( 0 , _{t 1} ) and _{h p 2} ( t ) > 0 for t ∈ ( _{t 1} , π / 2 ) , which also implies that _{H p 2} is decreasing on ( 0 , _{t 1} ) and increasing on ( _{t 1} , π / 2 ) . Therefore, [figure omitted; refer to PDF] that is, _{H p 2} ( t ) < 0 for t ∈ ( 0 , π / 2 ) .

It remains to prove that _{p 2} is the best possible constant. If there is a ^{p 2 [variant prime]} ∈ ( 0,1 ) with ^{p 2 [variant prime]} > _{p 2} such that the right inequality in (55) holds for t ∈ ( 0 , π / 2 ) , then, by the second assertion proved previously, we have _{H ^{p 2 [variant prime]}} ( π / ^{2 -} ) > 0 , 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 _{D p} ( t ) defined by [figure omitted; refer to PDF] for t ∈ ( 0 , π / 2 ) and p ∈ ( 0,1 ) . Arranging leads to [figure omitted; refer to PDF] As shown previously, _{H p} ( π / ^{2 -} ) < 0 for p ∈ ( 0 , _{p 2} ) and _{H p} ( π / ^{2 -} ) > 0 for p ∈ ( _{p 2} , 1 ) , which together with ln ... ( cos ... p t ) > ln ... ln ... ( cos ... ( π p / 2 ) ) and ln ... ( cos ... ( π p / 2 ) ) < 0 gives the desired result.

Lemma 7 reveals that the monotonicity of the first, second, and third members in (57) with respect to p on ( 0,1 ) due to [figure omitted; refer to PDF] Finally, we show that _{β p} is the best possible constant. It easily follows that [figure omitted; refer to PDF]

Thus, we complete the proof.

Remark 11.

Letting t = x / 2 and _{p 2} = 2 _{p 3} in Theorem 10 and then taking squares, we deduce that the two-side inequality [figure omitted; refer to PDF] holds for x ∈ ( 0 , π ) , where _{p 3} = _{p 2} / 2 [approximate] 0.31055 .

From the proof of Theorem 10, we clearly see that the constant 1 / 10 in (79) is the best possible constant, but _{p 3} = _{p 2} / 2 is not.

In [15, Theorems 1, 2, and 3], Yang proved that the chain of inequalities [figure omitted; refer to PDF] holds for t ∈ ( 0 , π / 2 ) with the best constants 1 / 5 and ^{p 0 *} [approximate] 0.45346 . The monotonicity of the function p ... ^{( cos ... p t ) 1 / ( 3 p 2 )} on ( 0,1 ) given in Lemma 7 and Remark 11 lead to the following.

Corollary 12.

For t ∈ ( 0 , π / 2 ) , the chain of inequalities [figure omitted; refer to PDF] holds with the best possible constants 2 / 10 [approximate] 0.63246 , _{p 2} [approximate] 0.6210901 , ^{p 0 *} [approximate] 0.45346 , 1 / 5 [approximate] 0.44721 and 1 / 10 [approximate] 0.31623 , and _{p 3} [approximate] 0.31055 .

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

Corollary 13.

For t ∈ ( 0 , π / 2 ) , one has [figure omitted; refer to PDF]

Proof.

The first, second, and third inequalities in (82) are due to Neuman [17, Theorem 1].

The fourth one in (82) is equivalent to [figure omitted; refer to PDF] which holds due to [figure omitted; refer to PDF] for t ∈ ( 0 , π / 2 ) .

The eighth one is derived from Neuman and Sándor [18, ( 2.5)].

The ninth one easily follows from [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] holds for t ∈ ( 0 , π / 2 ) . Clearly, the sixth and seventh inequalities in (82), that is, for t ∈ ( 0 , π / 2 ) , [figure omitted; refer to PDF] are a refinement of Sándor's inequality.

Remark 15.

Using the decreasing property of the function l defined by (83) proved in Corollary 13, we also get 1 = l ( ^{0 +} ) > l ( t ) > l ( π / ^{2 -} ) = 2 2 / 3 for t ∈ ( 0 , π / 2 ) , which can be rewritten as [figure omitted; refer to PDF] This in conjunction with (43) gives [figure omitted; refer to PDF] From [figure omitted; refer to PDF] we conclude that 1 / 3 and 1 / e are also the best possible constants.

Further, we conjecture that [figure omitted; refer to PDF] hold for t ∈ ( 0 , π / 2 ) , where all exponents are optimal.

Taking p = 1 / 2 , 1 / 3 , and ^{0 +} in (57), we get the following.

Corollary 16.

For t ∈ ( 0 , π / 2 ) , we have [figure omitted; refer to PDF] where _{α 1 / 2} = 2 / ln ... 2 [approximate] 2.8854 , _{α 1 / 3} = 2 / ( ln ... 4 - ln ... 3 ) [approximate] 6.9521 and _{β 1 / 2} = 2 2 3 ^{e - 1} [approximate] 0.92700 , _{β 1 / 3} = 64 ^{e - 1} / 27 [approximate] 0.87201 are the best possible constants.

Remark 17.

The inequalities connecting the first, fourth, and seventh members in (93) state that, for t ∈ ( 0 , π / 2 ) , [figure omitted; refer to PDF] which can be written as [figure omitted; refer to PDF] or [figure omitted; refer to PDF] It is easy to check that this double inequality is stronger than the new Redheffer-type one for tan t proved by Zhu and Sun [21, Theorem 3]; that is, for t ∈ ( 0 , π / 2 ) , [figure omitted; refer to PDF]

Remark 18.

Making use of the double inequalities [figure omitted; refer to PDF] for t ∈ ( 0 , π / 2 ) proved in [22] and [15, Corollary 3], respectively and taking into account (93) and (94), we easily obtain [figure omitted; refer to PDF]

3.3. The Sharp Bounds for ( sin t / t ) ^{e t cot t - 1}

In this subsection, we establish sharp inequalities between ( ^{t - 1} sin t ) ^{e t cot t - 1} and ^{( cos ... p t ) 1 / p 2} and prove the trigonometric version of inequalities (9) and (10). Employing Lemmas 6 and 7, we have the following.

Theorem 19.

For t ∈ ( 0 , π / 2 ) , the two-side inequality [figure omitted; refer to PDF] holds with the best possible constants 1 / 3 and _{p 4} [approximate] 0.5763247 , where _{p 4} is the unique root of the equation [figure omitted; refer to PDF] on ( 1 / 2,1 ) . Moreover, the inequalities [figure omitted; refer to PDF] hold for p ∈ [ 1 / 3 , 1 ] , where the exponents [figure omitted; refer to PDF] and the coefficients [figure omitted; refer to PDF] are the best possible constants. Also, the first member in (104) is decreasing with respect to p on ( 0,1 ) , while the third and fourth members are increasing with respect to p on ( 0,1 ) . The reverse of (104) holds if p ∈ ( 0,1 / 2 ] .

Proof.

For t ∈ ( 0 , π / 2 ) and p ∈ ( 0,1 ) , we define [figure omitted; refer to PDF] To prove the desired results, we need two assertions. The first is the limit relation [figure omitted; refer to PDF] which follows by expanding in power series [figure omitted; refer to PDF] The second one states that the equation _{J p} ( π / ^{2 -} ) = 0 , that is, (103), has a unique solution _{p 4} [approximate] 0.5763247 such that _{J p} ( π / 2 ) < 0 for p ∈ ( 0 , _{p 4} ) and _{J p} ( π / ^{2 -} ) > 0 for p ∈ ( _{p 4} , 1 ) . In fact, Lemma 7 implies that p ... _{J p} ( π / ^{2 -} ) is increasing on ( 0,1 ) , which in conjunction with the facts that [figure omitted; refer to PDF] indicates the second one. By using mathematical software, we find _{p 2} [approximate] 0.5763247 .

(i) Now we show that the first inequality in (102) holds for t ∈ ( 0 , π / 2 ) with the best constants 1 / 3 . In fact, the first inequality in (102) follows by Lemma 6. On the other hand, due to the decreasing property of p - 2 ln ... ( cos ... p t ) with respect to p on ( 0,1 ) , if there is a smaller p * ∈ ( 0,1 ) with p * < 1 / 3 such that the first inequality in (102) holds for t ∈ ( 0 , π / 2 ) , then there must be lim ... t [arrow right] 0 + t - 4 J p * ( t ) ...5; 0 , which by the relation (108) gives p * ...5; 1 / 3 . This yields a contradiction. Consequently, the constants 1 / 3 is optimal.

(ii) We next prove that the second inequality in (102) holds for t ∈ ( 0 , π / 2 ) , where _{p 4} is the best possible constant. We introduce an auxiliary function _{j p 4} defined on ( 0 , π / 2 ) by [figure omitted; refer to PDF] Expanding in power series leads to [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Therefore, we have [figure omitted; refer to PDF] Differentiation again yields [figure omitted; refer to PDF] We assert that ^{( t - 1 j _{p 4} [variant prime] ( t ) ) [variant prime]} > 0 for t ∈ ( 0 , π / 2 ) . It suffices to show that _{s n} > 0 for n ...5; 4 . In fact, _{s 4} = 3 ( 85 ^{p 4 6} - 3 ) > 0 , and _{s n} satisfies the recursive relation [figure omitted; refer to PDF] A direct check gives [ 64 × ^{2 4 n} - ( 36 ^{n 2} + 108 n + 113 ) ^{2 2 n} _{+ 4 ] n = 4} = 3907332 , [figure omitted; refer to PDF] due to ^{s 4 [variant prime][variant prime]} = 3907332 and ^{s n [variant prime][variant prime]} satisfies the recursive relation [figure omitted; refer to PDF] Hence, ^{s n [variant prime]} is decreasing for n ...5; 4 , and so [figure omitted; refer to PDF] which yields ^{p 4 2} - ^{s n [variant prime]} > ^{p 4 2} - 85 / 279 > 0 . From the recursive relation (116), we get _{s n} > 0 for n ...5; 4 , which proves that ^{( t - 1 j _{p 4} [variant prime] ( t ) ) [variant prime]} > 0 for t ∈ ( 0 , π / 2 ) . Therefore, we get [figure omitted; refer to PDF] Next, we divide the proof into two cases.

Case 1 ( ( ^{t - 1 j _{p 4} [variant prime]} ( t ) ) _{| t = π / ^{2 -}} < 0 ) . In this case, we clearly see that ^{t - 1 j _{p 4} [variant prime]} ( t ) < 0 for t ∈ ( 0 , π / 2 ) and ^{j _{p 4} [variant prime]} ( t ) < 0 for t ∈ ( 0 , π / 2 ) . Hence, _{j p 4} ( t ) < _{j p 4} ( ^{0 +} ) = ( 3 ^{p 4 2} - 1 ) / 9 < 0 , and so ^{J _{p 4} [variant prime]} ( t ) < 0 for t ∈ ( 0 , π / 2 ) , which reveals that _{J p 4} ( t ) < _{J p 4} ( ^{0 +} ) = 0 and _{J p 4} ( t ) > _{J p 4} ( π / ^{2 -} ) = 0 for t ∈ ( 0 , π / 2 ) , where _{J p 4} ( π / ^{2 -} ) = 0 due to _{p 4} being the unique root of (103). This is impossible.

Case 2 ( ( ^{t - 1 j _{p 4} [variant prime]} ( t ) ) _{| t = π / ^{2 -}} > 0 ) . In this case, we see that there is a _{t 2} ∈ ( 0 , π / 2 ) such that ^{t - 1 j _{p 4} [variant prime]} ( t ) < 0 for t ∈ ( 0 , _{t 2} ) and ^{t - 1 j _{p 4} [variant prime]} ( t ) > 0 for t ∈ ( _{t 2} , π / 2 ) . This indicates that _{j p 4} is decreasing on ( 0 , _{t 2} ) and increasing on ( _{t 2} , π / 2 ) . Thus, we have _{j p 4} ( t ) < _{j p 4} ( ^{0 +} ) = ( 3 ^{p 4 2} - 1 ) / 9 < 0 for t ∈ ( 0 , _{t 2} ) .

If _{j p 4} ( π / ^{2 -} ) < 0 , then _{j p 4} ( t ) < 0 for t ∈ ( 0 , π / 2 ) . Similar to Case 1, this also yields a contradiction.

If _{j p 4} ( π / ^{2 -} ) > 0 , then there is a _{t 3} ∈ ( _{t 2} , π / 2 ) such that _{j p 4} ( π / ^{2 -} ) < 0 for t ∈ ( 0 , _{t 3} ) and _{j p 4} ( π / ^{2 -} ) > 0 for t ∈ ( _{t 3} , π / 2 ) , which together with (111) shows that _{J p 4} is decreasing on ( 0 , _{t 3} ) and increasing on ( _{t 3} , π / 2 ) . Therefore, [figure omitted; refer to PDF] that is, _{J p 4} ( t ) < 0 for t ∈ ( 0 , π / 2 ) .

On the other hand, if there is a ^{p *} ∈ ( 0,1 ) with ^{p *} > _{p 4} such that the second inequality in (55) holds for t ∈ ( 0 , π / 2 ) , then by the second assertion proved previously, we have _{J ^{p *}} ( π / ^{2 -} ) > 0 , which leads to a contradiction. This proves that the constant _{p 4} 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 _{E p} ( t ) defined by [figure omitted; refer to PDF] for t ∈ ( 0 , π / 2 ) and p ∈ ( 0,1 ) . Simplifying leads to [figure omitted; refer to PDF] As shown previously, _{J p} ( π / ^{2 -} ) < 0 for p ∈ ( 0 , _{p 4} ) and _{J p} ( π / ^{2 -} ) > 0 for p ∈ ( _{p 4} , 1 ) , which in combination with ln ... ( cos ... p t ) > ln ... ( cos ... ( π p / 2 ) ) and ln ... ( cos ... ( π p / 2 ) ) < 0 gives the desired result.

Lemma 7 reveals the monotonicity of the first, second, and third members in (104) with respect to p on ( 0,1 ) due to [figure omitted; refer to PDF] Finally, we prove that _{β p} is the best possible constant. It can be deduced from [figure omitted; refer to PDF]

Thus, the proof is complete.

We note that (102) can be written as [figure omitted; refer to PDF] Making use of the monotonicity of the function p ... ^{( cos ... p t ) 1 / ( 3 p 2 )} on ( 0,1 ) given in Lemma 7 together with Corollary 12 and Theorem 19, we obtain the following.

Corollary 20.

For t ∈ ( 0 , π / 2 ) , the chain of inequalities [figure omitted; refer to PDF] holds, where 2 / 10 [approximate] 0.63246 , _{p 2} [approximate] 0.6210901 , 1 / 3 [approximate] 0.57735 , _{p 4} [approximate] 0.5763247 , ^{p 0 *} [approximate] 0.45346 , 1 / 5 [approximate] 0.44721 , and 1 / 10 [approximate] 0.31623 are the best possible constants, and _{p 3} [approximate] 0.31055 .

Remark 21.

From the above corollary, we clearly see that [figure omitted; refer to PDF] for t ∈ ( 0 , π / 2 ) . The relation connecting the first, third, and fourth members in (128) can be written as [figure omitted; refer to PDF]

Taking p = 1 / 2 , ^{0 +} in Theorem 19, we have the following.

Corollary 22.

For t ∈ ( 0 , π / 2 ) , the inequalities [figure omitted; refer to PDF] hold, where the exponents _{γ 1 / 2} = 2 ( ln ... ( π e / 2 ) ) / ln ... 2 [approximate] 4.1884 and 4 and the coefficients 1 and 8 / ( π e ) [approximate] 0.93680 are the best possible constants.

Theorem 23.

For t ∈ ( 0 , π / 2 ) , we have [figure omitted; refer to PDF] where e ( π - 2 ) / π , 1 , ( ^{e - 1} + 2 / π ) , 2 / π - ^{e - 1} , and 1 / 3 are the best possible constants.

Proof.

(i) We first prove (132). For this purpose, let us define [figure omitted; refer to PDF] Differentiating k ( t ) gives [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Using double angle formula and Lemma 4, we have [figure omitted; refer to PDF] Hence, ^{k [variant prime]} ( t ) > 0 for t ∈ ( 0 , π / 2 ) , and so [figure omitted; refer to PDF] which implies the desired inequalities.

(ii) Now, we prove (133). Differentiation leads to [figure omitted; refer to PDF] where the inequality holds for t ∈ ( 0 , π / 2 ) due to (88). Therefore, [figure omitted; refer to PDF] which deduces (133).

(iii) Similarly, we have [figure omitted; refer to PDF] which gives [figure omitted; refer to PDF]

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

Corollary 24.

For t ∈ ( 0 , π / 2 ) , we have [figure omitted; refer to PDF]

3.4. The Third Sharp Bounds for ^{e t cot t - 1}

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

Theorem 25.

Let t ∈ ( 0 , π / 2 ) . Then the following statements are true:

(i) if p ...5; 6 / 5 , then the two-side inequality [figure omitted; refer to PDF] holds if and only if α ...5; 1 - ^{e - p} and β ...4; 2 / 3 ;

(ii) if 0 < p ...4; 1 , then the double inequality (145) holds if and only if α ...5; 2 / 3 and β ...4; 1 - ^{e - p} ;

(iii): if p < 0 , then the double inequality (145) holds if and only if α ...4; 0 and β ...5; 2 / 3 ;

(iv) the double inequality [figure omitted; refer to PDF]

holds if and only if p ...4; ln ... 3 and q ...5; 6 / 5 , where _{M p} ( x , y ; w ) ( w ∈ ( 0,1 ) ) is the weighted power mean of order r of x and y defined by [figure omitted; refer to PDF]

Proof.

For t ∈ ( 0 , π / 2 ) and p ...0; 0 , we define [figure omitted; refer to PDF] Since _{u 1} ( ^{0 +} ) = _{u 2} ( ^{0 +} ) = 0 , u ( t ) can be written as [figure omitted; refer to PDF] Differentiation gives [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Clearly, if we prove that ^{u 3 [variant prime]} ( t ) > 0 for p ...5; 6 / 5 and ^{u 3 [variant prime]} ( t ) < 0 for p ...4; 1 with p ...0; 0 , then, by Lemma 2, we know that u is increasing if p ...5; 6 / 5 and decreasing if p ...4; 1 with p ...0; 0 , and [figure omitted; refer to PDF] which yield the first, second, and third results in this theorem.

Now, we show that ^{u 3 [variant prime]} ( t ) > 0 if p ...5; 6 / 5 and ^{u 3 [variant prime]} ( t ) < 0 if p ...4; 1 with p ...0; 0 . Simple computations lead to [figure omitted; refer to PDF] for t ∈ ( 0 , π / 2 ) . Using (15)-(17), we have [figure omitted; refer to PDF] By Lemma 3, in order to prove the monotonicity of _{u 5} ( t ) / _{u 4} ( t ) , it suffices to get the monotonicity of _{a n} / _{b n} . Note that [figure omitted; refer to PDF] Differentiating c ( x ) , we get [figure omitted; refer to PDF] for x ...5; 2 . The function t ... _{u 5} ( t ) / _{u 4} ( t ) is decreasing on ( 0 , π / 2 ) , and we conclude that [figure omitted; refer to PDF] Thus, ^{u 3 [variant prime]} ( t ) > 0 if p ...5; 6 / 5 and ^{u 3 [variant prime]} ( t ) < 0 if p ...4; 1 with p ...0; 0 .

Finally, we prove the fourth result. The first part implies that the right-hand side inequality in (146) holds if q ...5; 6 / 5 . While the necessity can be obtained from the following limit relation: [figure omitted; refer to PDF] in fact, power series expansion leads to [figure omitted; refer to PDF]

Now, we prove that the left-hand side inequality holds if and only if p ...4; ln ... 3 . The necessity follows easily from [figure omitted; refer to PDF] Next, we deal with the sufficiency. We divide the proof into two cases.

Case 1 ( p ...4; 1 ) . The sufficiency follows immediately from the second and third results proved previously.

Case 2 ( 1 < p ...4; ln ... 3 ) . It was proved previously that the function t ... _{u 5} ( t ) / _{u 4} ( t ) is decreasing on ( 0 , π / 2 ) , and so the function t ... ( p - _{u 5} ( t ) / _{u 4} ( t ) ) : = _{u 6} ( t ) is increasing on the same interval. The monotonicity _{u 6} ( t ) together with [figure omitted; refer to PDF] leads to the conclusion that there exists unique _{t 0} ∈ ( 0 , π / 2 ) such that _{u 6} ( t ) < 0 for t ∈ ( 0 , _{t 0} ) and _{u 6} ( t ) > 0 for t ∈ ( _{t 0} , π / 2 ) ; then, from (151), we know that _{u 3} is decreasing on ( 0 , _{t 0} ) and increasing on ( _{t 0} , π / 2 ) . It follows from Lemma 2 that u is decreasing on ( 0 , _{t 0} ) , and so we have [figure omitted; refer to PDF] which can be rewritten as [figure omitted; refer to PDF] On the other hand, Lemma 2 also implies that [figure omitted; refer to PDF] is increasing on ( _{t 0} , π / 2 ) . Therefore, [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] 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 _{t 0} satisfies (164), for t ∈ ( _{t 0} , π / 2 ) , we have [figure omitted; refer to PDF] where the last inequality holds due to p ∈ ( 1 , ln ... 3 ] and t ∈ ( _{t 0} , π / 2 ) .

Thus, the proof is finished.

4. Some Corresponding Inequalities for Means

The Schwab-Borchardt mean of two numbers a ...5; 0 and b > 0 is defined by [figure omitted; refer to PDF] (see [23, Theorem 8.4], [24, ( 2.3 ) ], and [25, ( 1.1 ) ]). It is clear that SB ( a , b ) is not symmetric in its variables and is a homogeneous function of degree 1 in a and b . 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 b ...5; a > 0 and p , q ∈ [ - 2,2 ] such that 0 ...4; p + q ...4; 3 , and let S ~ ( p , q , t ) be defined by [figure omitted; refer to PDF] Then _{S p , q} ( a , b ) defined by [figure omitted; refer to PDF] is called a two-parameter sine mean of a and b .

In particular, for b ...5; a > 0 , [figure omitted; refer to PDF] are means of a and b . Similarly, according to the definition of two-parameter cosine mean (see [19, Definition 4.2]), [figure omitted; refer to PDF] is also a mean of a and b , where _{U p} ( t ) is defined by (34).

Further, we have the following.

Proposition 27.

For b ...5; a > 0 and α ∈ ( 0,1 ] , the function [figure omitted; refer to PDF] is also a mean of a and b , where _{V p} ( t ) is defined by (35).

Proof.

It suffices to prove that the double inequality [figure omitted; refer to PDF] holds for b > a > 0 , which is equivalent to [figure omitted; refer to PDF] where t = ( arccos ( a / b ) ) ∈ ( 0 , π / 2 ) .

Using the decreasing property proved in Lemma 7, we see that [figure omitted; refer to PDF] which proves the assertion.

If we replace t by arccos ( a / b ) and then multiply b or ^{b λ} for suitable λ 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 ^{10 [variant prime]} . For b ...5; a > 0 , the two-side inequality [figure omitted; refer to PDF] holds with the best possible constants 2 / 10 and _{p 2} [approximate] 0.6210901 , where _{p 2} is the unique solution of (56) on ( 1 / 2,1 ) .

Theorem ^{19 [variant prime]} . For b ...5; a > 0 , the two-side inequality [figure omitted; refer to PDF] holds with the best constants 1 / 3 and _{p 4} [approximate] 0.5763247 , where _{p 4} is the unique root of (103) on ( 1 / 2,1 ) .

Theorem ^{25 [variant prime]} . Let b ...5; a > 0 .Then the following statements are true:

(i) if p ...5; 6 / 5 , then the two-side inequality [figure omitted; refer to PDF]

holds if and only if α ...5; 1 - ^{e - p} and β ...4; 2 / 3 ;

(ii) if 0 < p ...4; 1 , then the double inequality (180) holds if and only if α ...5; 2 / 3 and β ...4; 1 - ^{e - p} ;

(iii): if p < 0 , then the double inequality (180) holds if and only if α ...4; 0 and β ...5; 2 / 3 ;

(iv) the double inequality [figure omitted; refer to PDF]

holds if and only if p ...4; l n 3 and q ...5; 6 / 5 , where the left hand side in (181) is defined as ^{a 2 / 3 b 1 / 3} if p = 0 .

Similar to S B ( a , b ) , these bivariate means mentioned previously are not symmetric in their variables and are homogeneous of degree 1 in a and b . But they can generate more symmetric means by making certain substitutions; for example, Neuman and Sándor [25, ( 1.1 ) ] proved that SB ( G , A ) = P , SB ( A , Q ) = T , where Q , A , G , P , and T denote the quadratic, arithmetic, geometric, first, and second Seiffert means [28, 29] of a and b given by [figure omitted; refer to PDF] respectively. In same way, we have [figure omitted; refer to PDF] which is a Sándor mean introduced in [20, page 82], [30]. Also, we get [figure omitted; refer to PDF] which is also a new mean, and it satisfies the double inequality A < B < Q .

There are many inequalities involving means Q , A , G , P , and T ; we quote [15, 20, 25, 27, 31-44]. Inequalities for Sándor's mean X 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 t = arccos ( a / b ) , where b ...5; a > 0 .

Step 2.

Put ( a , b ) = ( m ( x , y ) , M ( x , y ) ) , where m ( x , y ) , M ( x , y ) are means of positive numbers x and y , and m ( x , y ) ...4; M ( x , y ) for all x , y > 0 .

Let ( m , M ) = ( G , A ) and ( m , M ) = ( A , Q ) . Then the following variable substitutions follows from Steps 1 and 2.

(i) Substitution 1: t = arccos ( G / A ) . Then [figure omitted; refer to PDF]

(ii) Substitution 2: t = arccos ( A / Q ) . Then [figure omitted; refer to PDF] For simplicity in expressions, we only select the functions involving ( sin t ) / t , cos ... t , and cos ... ( t / 2 ) in a chain of inequalities given in Section 3.

The following follows from (88).

Proposition 28.

For b ...5; a > 0 , the inequalities [figure omitted; refer to PDF] hold. Moreover, replacing ( a , b ) by ( G , A ) , we have [figure omitted; refer to PDF] replacing ( a , b ) by ( A , Q ) , we get [figure omitted; refer to PDF]

Remark 29.

The second inequalities in (188) and (189) are due to Sándor [31, 33], while the one connecting P and A X first appeared in [20, page 82, ( 2.6 ) ].

From inequalities (90), we have the following.

Proposition 30.

For b ...5; a > 0 , the double inequality [figure omitted; refer to PDF] is valid, where 3 and e are the best possible constants. Moreover, replacing ( a , b ) by ( G , A ) and ( A , Q ) , we get [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] proved by Sándor [20, page 89, ( 2.14 ) ].

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

Proposition 32.

For b ...5; a > 0 , the inequalities [figure omitted; refer to PDF] hold true with the best possible exponents and coefficients. Moreover, replacing ( a , b ) by ( G , A ) and ( A , Q ) , we have [figure omitted; refer to PDF]

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 b ...5; a > 0 , the inequalities [figure omitted; refer to PDF] hold, where the exponents _{γ 1 / 2} / 4 = ( ln ... ( π e / 2 ) ) / ln ... 4 [approximate] 1.0471 and 1 and the coefficients 8 / ( π e ) [approximate] 0.96788 and 1 are the best possible constants. Moreover, replacing ( a , b ) by ( G , A ) and ( A , Q ) , we have [figure omitted; refer to PDF]

Inequalities (134) lead to the following.

Proposition 35.

For b ...5; a > 0 , the sharp inequalities [figure omitted; refer to PDF] hold true. Moreover, replacing ( a , b ) by ( G , A ) , ( A , Q ) , we have [figure omitted; refer to PDF]

Inequalities (144) lead to the following conclusion.

Proposition 36.

For b ...5; a > 0 , the inequalities [figure omitted; refer to PDF] are valid, where ^{e - 1} + 2 / π [approximate] 1.0045 is the best possible constant. In particular, replacing ( a , b ) by ( G , A ) and ( A , Q ) , we get [figure omitted; refer to PDF]

Remark 37.

The first inequality in (202) was established by Sándor in [20, page 87, ( 2.2 ) ].

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.

Conflict of Interests

The authors declare that they have no competing interests.

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