A two-point-Padé-approximant-based method for bounding some trigonometric functions.

Inequalities are frequently used for solving practical engineering problem. There are two key issues of bounding inequalities; one is to find the bounds, and the other is to prove the bounds. This paper takes Wilker type inequalities as an example, presents a two-point-Padé-approximant-based method for finding the bounds, and it also provides a method to prove the bounds in a new way. It not only recovers the estimates in Mortici's method, but it also provides new improvements of estimates obtained from prevailing methods. In principle, it can be applied for other inequalities.

Introduction

The Wilker inequality, which involves the trigonometric function has been discussed in the recent past; see also [2, 3, 6–9, 11–15, 17–23] and the references therein, such as the following ones in [14, 18]: where , , , .

Recently, Nenezić, Malešević and Mortici provided inequalities within the extended interval [15], e.g., Eq. (7) extends both Eq. (4) and Eq. (5), while Eq. (6) extends the left side of Eq. (3). We have where with .

In this paper, we consider instead of , which is bounded for . Firstly, we present a two-point-Padé approximant-based method [1] to find the two bounding functions such that where and are unknown polynomials to be determined. Note that , from Eq. (10), we obtain Secondly, we also provide a new way for proving it.

The two-point-Padé approximant-based method and examples

Given an interval . From Eq. (9), let where , and are the unknowns to be determined, and ; so there are and unknowns in and in Eq. (9), respectively. Let and . For the sake of convenience, we introduce Theorem 3.5.1 in Page 67, Chap. 3.5 of [4] as follows.

Theorem 1

Let be distinct points in , and be integers ≥0. Let . Suppose that is a polynomial of degree N such that , , . Then there exists such that .

We introduce the following constraints: where and . By selecting suitable k and , we can find constraints for determining ; similarly, by selecting suitable l and , we can find constraints for determining . Combining Theorem 1 with Eq. (13), there exists , such that From Eq. (14), if , , we have , where or ; similarly, if , , we have . Based on the above observations, one may find the bounding functions in the above way.

We will show three examples which recover or refine previous Wilker type inequalities, including Eq. (2), Eq. (6) and Eq. (7), where is a unknown coefficient to be determined by interpolation constraints.

Example 1

Let and , and , . It can be verified that , where , . By applying the constraints and , we obtain and , respectively, which recovers Eq. (2).

Example 2

Let and , and , . It can be verified that , where , . By applying the constraints , , we obtain , and , which recovers the left side of Eq. (6). By applying the constraints , and , we obtain , and , which recovers the right side of Eq. (6).

Example 3

Let , , and , . It can be verified that , where , . By applying the constraints and , we obtain and , which recovers the left side of Eq. (7). By applying the constraints , , we obtain , and , which recovers the right side of Eq. (7).

Results

This section finds other two bounding functions and to improve the bounds of Eq. (6) and Eq. (7). Combining Eq. (12) with Eq. (13), by setting , , , , and , we obtain and in Eq. (10) as where In principle, more bounds can be found by setting different parameters in Eq. (12) and Eq. (13). The main result is as follows.

Theorem 2

We have , .

Proof

(1) Firstly, we give the bounds of , and . Let , , , , , , where , , , , and are polynomials of degree 12, 12, 13, 13, 15 and 15, respectively. By introducing the following constraints: we can obtain , , , , , , where , , , , , , , , , , , , , , .

Combining Theorem 1 with Eq. (15), there exists , , such that

So for , we have i.e., , and .

(2) Secondly, we prove that , , which means that .

Note that and are polynomials of degree , , polynomials , , , , and are of degree 12, 12, 13, 13, 15 and 15, respectively, by using Maple software, , we obtain Combining Eq. (17) with Eq. (16), we have where and Note that , we have , . It leads to and , .

(3) Finally, we prove that , , which means that . Combining Eq. (17) with Eq. (16), we have where and Note that , we have . So we have and , .

From the above discussions, we have completed the proof. □

Discussions and conclusions

In principle, one can prove that , in a similar way, where and , , are two bounding functions in Eq. (6) and Eq. (7), respectively. The maximum errors between and its different bounds are listed in Table 1. It shows that the bounds in this paper achieve a much better approximation than those of the bounds in Eq. (6) and Eq. (7).

Table 1

Maximum errors between and its different bounds

Bounds	L_1(x)	R_1(x)	L_2(x)	R_2(x)	L(x)	R(x)
Error	2.8e−2	2.5e−4	2.09e−3	1.7e−3	1.37e−5	4.89e−6

The new method can be applied to refine the Becker–Stark inequality, which is studied in [5, 16, 24] and is known as Zhu [24] refined it as while it is refined in [16] as follows: where .

By applying the method in Sect. 2 and using the form , one obtains the resulting bounds, and , where and , such that By using the Maple software, , it can be verified that , , and . So the bounds and achieve a better approximation than those results in both [24] and [16].