Refining trigonometric inequalities by using Padé approximant.

A two-point Padé approximant method is presented for refining some remarkable trigonometric inequalities including the Jordan inequality, Kober inequality, Becker-Stark inequality, and Wu-Srivastava inequality. Simple proofs are provided. It shows to achieve better approximation results than those of prevailing methods.

Introduction

Trigonometric inequalities have caused interest of a lot of researchers, they analyzed the Wilker inequality [6–11, 14, 16–19], Jordan inequality [3, 5, 15, 20, 21], Shafer–Fink inequality [12], Becker–Stark inequalities [13], and so on.

Recently, Bercu provided a Padé-approximant-based method and obtained the following inequalities [2]. where , and .

In this paper, we present a two-point Padé-approximant-based method [1] for refining the rational bounds of several trigonometric inequalities, and also provide a method for proving the refined bounds. By applying the new method to and , we refine the bounds of Eq. (1) ∼ (2), for , see also Theorems 3.1 and 3.2. Applied to and , it not only provides refined two-sided bounds with better approximation effect for Eq. (3) ∼ (4), but also extends the interval to the interval , see also the theorems and remarks in Sect. 3.

Find bounds by using two-point Padé approximant

Given a bounded smooth function , , let be a rational polynomial interpolating derivatives of at two points and such that where . There are unknowns in Eq. (5). By selecting suitable values of k and l, we have that Eq. (5) consists of linear equations in the unknown variables and , and the interpolation polynomial can be determined by solving Eq. (5).

We give two examples. Without loss of generality, let .

Example 1

Let . By setting , , , and and introducing the following constraints we obtain that where , , , . It can be verified that . From Eq. (6), , there exists such that [4] Note that and , . Combining with Eq. (8), one obtains that

Example 2

Let . By setting , , , and and introducing the following constraints we obtain that where , , , . It can be verified that . From Eq. (10), , there exists , such that [4] Note that and , . Combining with Eq. (12), one obtains that

Main results

The main results are as follows.

Theorem 3.1

For all , we have that

Proof

Eq. (14) is equivalent to It is well known that , Combining with Eq. (16), we have that which is just Eq. (15). So we have completed the proof of Eq. (14). □

Theorem 3.2

For all , we have that

Eq. (17) is equivalent to It is well known that Combining with Eq. (19), we have that Thus, we have completed the proof of both Eq. (18) and Eq. (17). □

Theorem 3.3

For all , we have that where and .

Eq. (21) is equivalent to It can be verified that Combining with Eq. (23), we have that Let , . On the other hand, it can be verified that, , Combining Eq. (23) with Eq. (25), we have that where , . Combining Eq. (26) with , we obtain that Combining Eq. (24) with Eq. (27), we have completed the proof of both Eq. (22) and Eq. (21). □

From Theorems 3.1, 3.2, and 3.3, we directly obtain the following theorem.

Theorem 3.4

We have that

Discussion and conclusions

Firstly, we compare the results of between in [2] and in this paper, . It can be verified that and , , we have that

Secondly, we compare the approximation results of between previous and present , . It can be verified that and , , we have that

Thirdly, we compare the approximation results of , which also shows that this paper achieves a much better result. It can be verified that , However, note that the denominator of is , which has a real root ≈1.5701 within the interval Γ, and we have . It can be verified that , where . By using the Maple software, has six real roots , and , we have that