Sharp inequalities for tangent function with applications.

In the article, we present new bounds for the function [Formula: see text] on the interval [Formula: see text] and find sharp estimations for the Sine integral and the Catalan constant based on a new monotonicity criterion for the quotient of power series, which refine the Redheffer and Becker-Stark type inequalities for tangent function.

Introduction

The study of this paper is concerned with the following inequality: which was posted by Redheffer in [1] and was proved by Williams [2]. Recently, Zhu and Sun [3] extended the Redheffer inequality (1.1) to the tangent function, and they established the following inequalities: with the best exponents and 1. Zhu [4] further refined the double inequality It is worth noting that Becker and Stark [5] in 1978 showed the double inequality where 8 and are the best constants. Later, Zhu and Hua [6] gave a general refinement of the Becker-Stark inequalities (1.4) by the power series expansion of the tangent function in terms of the Bernoulli numbers. In particular, they proved that for the double inequality holds with the best constants and ; also see [7]. Chen and Cheung [8] further presented an improvement of the left hand side inequality in (1.4), which states that holds for with the best exponents and (also cf. [9]). Another improvement involving the left hand side one in (1.4) was made in [10] by Nishizawa. Very recently, Bhayo and Sándor [11], Corollary 3, again proved the Becker-Stark inequalities (1.4) by using Redheffer inequality (1.1), which reveals the implicit relation between Redheffer’s and Becker-Stark’s inequalities. They in [11], Corollaries 2, also stated that for we have It is an important observation that Yang et al. [12], (93), in 2014 considered the bounds for function and established a number of inequalities for trigonometric functions. In particular, they in [12], Corollary 16, showed that for which can be written as

Inspired by these results mentioned above, the aim of this paper is to determine the best bounds for in terms of on , that is to say, we will determine the best parameters such that the double inequality holds for all . Inequalities (1.10) also can be rewritten as which offers a new type of bounds being different of the previous papers for the tangent function.

Some useful lemmas

In order to prove the main Theorem 1 in the next section, we need some preliminary lemmas. To this end, we first introduce a useful auxiliary function . For , let f and g be differentiable on and on . Then the function is defined by The function has been investigated with some well properties in [13], Properties 1, 2, which plays an important role in the proof of a monotonicity criterion for the quotient of power series; also see [14].

Lemma 1

[14], Theorem 2.1, [15], Lemma 3.1, and [16], Lemma 1.1

Let and be two real power series converging on and for all k. Suppose that, for certain , the non-constant sequence is increasing (resp. decreasing) for and decreasing (resp. increasing) for . Then the function is strictly increasing (resp. decreasing) on if and only if . Moreover, if , then there exists such that the function is strictly increasing (resp. decreasing) on and strictly decreasing (resp. increasing) on .

Lemma 2

[17], p. 75

Let . Then where is the Bernoulli number.

Lemma 3

[18]

Let and be the even-indexed Bernoulli numbers. Then Consequently, we have

Lemma 4

Let the function g be defined on by Then g is strictly increasing from onto and decreasing from onto .

Proof

To avoid complicated calculations, we here make use of Lemmas 1, 2 and 3 to prove this lemma. For this purpose, we write as then applying Lemma 2 yields where We now prove that the sequence is increasing for and decreasing for . A simple check yields and it remains to show that for . Indeed, we have Then by Lemma 3, we get where the inequality holds due to for . This proves the piecewise monotonicity of .

According to Lemma 1, we also have to check that and . In fact, we have then Lemma 1 leads to the result that there is a unique such that g is increasing on and decreasing on . Note that we clearly see that the unique . A simple computation yields which completes the proof. □

Remark 1

If we use an ordinary method to prove the piecewise monotonicity of g, then it is very troublesome. For example, a direct computation yields then differentiating gives As a result, there are various approaches to showing the piecewise monotonicity of on , but it seems to be difficult. It thus can be seen that our method used previously is relatively easy.

Lemma 5

For , let and be respectively defined on by (1.9) and Then and are strictly decreasing and increasing on , respectively. Moreover, there is a unique such that for and for , where is the unique solution of the equation on .

Let . Logarithmic differentiation yields where

Differentiation again leads to for , which means that . These together with and show that and are strictly decreasing and increasing on , respectively.

Note the increasing property of on and which implies that there is a unique such that for and for . Solving the equation for p gives . The proof is finished. □

Main results

This section is devoted to stating and proving the main results concerning some inequalities for the tangent function. More precisely, we have the following.

Theorem 1

For , let and be defined on by (1.9). When , the double inequality (3.2) still holds for all . In particular, when , we have with the best constants 1 and .

If , then the function is strictly decreasing on the interval . Consequently, for all with the best coefficients 1 and defined by (2.6).

If , then the function is strictly increasing on , and therefore, for all , with the best coefficients 1 and defined by (2.6).

If , then there is a such that the function is strictly increasing on and decreasing on , and hence, for all , with where is the unique solution of the equation on .

Let Differentiation yields where is defined by (2.5).

Noticing that for and for and , we easily see that, for all , As shown in Lemma 4, the function g is strictly increasing from onto . We are now in a position to distinguish three cases to prove the required result.

Case 1: . Then we obtain for , which means that f is strictly decreasing on . Consequently, we can deduce the following observation: which is equivalent to the double inequality (3.1) holding for all with the best coefficients 1 and .

Case 2: . Similarly, we have for , which implies that the double inequality (3.2) holds for all with the best coefficients 1 and .

Case 3: . Since is strictly decreasing on with we find that there is such that for and for . This indicates that f is strictly increasing on and decreasing on . Therefore, we deduce that for all , that is, (3.3) holds for all , where .

When , by Lemma 5 we have , and it follows that that is, the double inequality (3.2) still holds for .

In particular, for , solving equation (3.4) for t yields , and hence . Thus we complete the proof. □

Taking , respectively. Then by part (i) of Theorem 1 and the monotonicity of and given in Lemma 5, we immediately obtain the following conclusion.

Corollary 1

For , the inequalities holds with the best coefficients

Likewise, taking , respectively, we have the following.

Corollary 2

For , the inequalities hold with the best coefficients

Theorem 2

Let . Then the double inequality holds for all if and only if and , where is defined in Lemma 5.

Clearly, the sufficiency easily follows by Theorem 1. The necessary condition for the right hand side inequality in (3.10) to hold for follows from the limit relation The necessary condition for the left hand side inequality in (3.10) to hold for can be obtained from the inequality It follows from Lemma 5 that , which completes the proof. □

Comparisons and remarks

By Theorem 2, we have where .

We denote the lower bounds for given in the inequalities (4.1), (1.2), (1.3), (1.4), (1.5), and (1.7), respectively, by

Proposition 1

The comparison inequalities hold for . Moreover, , and are not comparable with each other for all .

(i) We first prove for . Differentiation yields which shows that is increasing on and decreasing on , where Then we conclude that with due to satisfying shown in Lemma 5.

(ii) The second inequality directly follows from for .

(iii) The third one is deduced by for .

(iv) Finally, we prove that , and are not comparable with each other for all . Simple computations yield

This completes the proof. □

Remark 2

From the above proposition we see that the sharp lower bound in (4.1) is superior to those ones given in (1.2), (1.3), (1.4), (1.5), and (1.7).

Remark 3

Analogously, by comparing the limits at and , we find the sharp upper bound in (4.1) is not comparable with those ones given in (1.2), (1.3), (1.4), (1.5), and (1.7). Here we omit all the details.

Remark 4

We claim that the result stated in Theorem 2 is stronger than the inequality (1.8), that is, for , we have the inequalities Indeed, the right hand side for this inequality in (4.3) follows from Corollary 1, while the left hand side one is the inequality connecting the first and third bounds in (4.2).

Remark 5

Lemma 4 tells us that Then from equation (3.7) we find that for for when , and so . This gives the following inequality: for all , which can be stated as the following proposition.

Proposition 2

For all , we have

Remark 6

The inequality is true due to Cusa and Huygens’ paper (see, e.g. [19]), which is now known as Cusa’s inequality (see e.g. [8, 20–23]). Some refinements and generalizations of Cusa’s inequality can be found in [8, 21, 22, 24–29]. Now by letting and simplifying, inequalities (4.4) and (4.5) can be written as which give stronger versions of Cusa’s inequality.

Proposition 3

We have Moreover, the two double inequalities are sharp.

Remark 7

In [12], Corollary 12, Yang et al. proved that, for , Then by inequalities (3.10) for and , we obtain for . Further, the right hand side inequalities can be improved as follows.

Proposition 4

The inequalities hold for with the best constants , and

Let where . Differentiation yields Expanding in power series leads to This indicates that for , which proves the second and third inequalities of (4.8). Considering the limit it is seen that and are the best possible constants.

The first and fourth ones are derived from the decreasing property of for proved in Theorem 1 for , and then and are also the best. This completes the proof. □

Applications

In this section, we give some precise estimations for the Sine integral and Catalan constant. The Sine integral is defined by There are many interesting results concerning the Sine integral; see [26, 30–33] and the references therein. Now we shall give more accurate estimations.

Proposition 5

For , we have In particular, we have

Indeed, integrating both sides over for double inequality (4.6) easily yields (5.1). Direct computations give the approximation values of for . □

Note that and We are now in the position to evaluate the integral for .

Proposition 6

Let . Then, for , we have where The double inequality (5.3) is reversed for . In particular, we have

By the proof of Theorem 1 we see that the function is strictly decreasing on if and increasing on if . Then, for , we have, for , that is, which is the reverse for .

Integrating both sides over gives, for , Combining with equation (5.2) gives the double inequality (5.3) for .

Moreover, it is clear that the double inequality (5.3) is reversed if .

Letting in (5.3) yields (5.4), which completes the proof. □

Remark 8

Taking and in the double inequality (5.3) and computing give Then we obtain Clearly, the absolute errors of the two approximations are less than 0.0007 and 0.00007.

It is well known that the Catalan constant appearing in [34-36] is a famous mysterious constant appearing in many places in mathematics and physics. Its integral representations [37] include the following: Now, by using the third integral representation for G and (5.6), we easily obtain a very accurate approximation for G, the absolute error of which is less than 0.00015.

Proposition 7

We have

Remark 9

Clearly, the above estimate for G is superior to Yang’s presentation in [26], Proposition 4, [33], Remark 4.2.

Conclusions

Rather than using classical approaches, we in this paper presented the new upper and lower bounds of on the interval by way of the monotonicity criterion for the quotient of power series. Our conclusions have not only refined the Redheffer and Becker-Stark type inequalities concerning the tangent function, but they also showed some more precise estimations to the Sine integral and the Catalan constant. More precisely, our conclusion is that the sharp lower bound of is superior to all given results as showed by Proposition 1 in Section 4, although its sharp upper bound is not comparable with those given ones. In addition, we also derived a stronger version of Cusa’s inequality, and a very accurate approximation of the Catalan constant with the absolute error being less than 0.00015.