Journal of Mathematics
Volume 2020 (2020), Article ID 2753691, 7 pages
https://doi.org/10.1155/2020/2753691

Research Article

New Refinements and Improvements of Some Trigonometric Inequalities Based on Padé Approximant

Lina Zhang¹ and Xuesi Ma²

¹School of Computer Science and Technology, Henan Polytechnic University, Jiaozuo, Henan 454000, China
²School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, China

Correspondence should be addressed to Xuesi Ma; maxuesi@hpu.edu.cn

Received 26 March 2020; Accepted 6 May 2020; Published 19 May 2020

Academic Editor: Dimitri Mugnai

Copyright © 2020 Lina Zhang and Xuesi Ma. 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.

Abstract.

A multiple-point Padé approximant method is presented for approximating and bounding some trigonometric functions in this paper. We give new refinements and improvements of some trigonometric inequalities including Jordan’s inequality, Kober’s inequality, and Becker-Stark’s inequality. The analysis results show that our conclusions are better than the previous conclusions.

1. Introduction

Inequalities involving trigonometric functions are used in many areas of pure and applied mathematics. Trigonometric inequalities have attracted many researchers. Many improvements of Jordan’ inequality [1–11], Kober’s inequality [12–16], and Becker-Stark’s inequality [4, 17, 18] have been obtained.

Recently, Bercu presented a Padé approximant method [19] and obtained the following inequalities:

Zhang et al. [20] gave the improvements of inequalities (1)–(3):

where

In this paper, we present a multiple-point Padé approximant method for approximating and bounding some trigonometric functions. New refinements and improvements of related trigonometric inequalities are obtained, including Jordan’s inequality, Kober’s inequality, and Becker-Stark’s inequality. We introduce the concept of maximum error to compare our results with the previous methods. The results show that our bounds are tighter than the previous conclusions.

2. Multiple-Point Padé Approximant Method

The Padé approximant has been studied in many literature studies [19, 21–24]. In particular, Bercu et al. presented good results of several trigonometric inequalities using the Padé approximant. In this section, we present a multiple-point Padé approximant method. Given a bounded smooth function , let

be a rational polynomial interpolating of at multiple points such that

where and and are two given integers. There are unknowns in equation (9), and , . By selecting suitable values of , we can obtain the polynomial by solving equation (9).

The general Padé approximant method is a special case of the multiple-point Padé approximant. Here, we just need to consider one point. If can be written as a formal power series , where the coefficients , are constant. Taylor’s expansion is one of the most common ways to get a power series of a function. The Padé approximant of degree of the function is determined by

The Padé approximant is considered the “best” approximation of a function by a rational function of a given degree. The rational approximation is also good for series with alternation terms and poor polynomial convergence. This is our motivation of using the Padé approximant to approximate trigonometric functions and improve these trigonometric inequalities. Different values of and will affect the approximate performance. By selecting suitable values of and , we can obtain the “best” approximant. Let , and we can obtain a simple result. The result is a special case of (10).

It is well known that

where

for , and are Bernoulli’s numbers.

Using the Padé approximant and equation (11), we obtain a better approximation of tangent function. Here, we need to pay attention to the value of in formula (10). Let , and is given in (12); we can obtain the Padé approximant of . In the same way, we can also obtain the Padé approximants of other trigonometric functions.

Table 1 gives the comparison between the Padé approximant and the Taylor series expansion of tangent function. It is easy to see that the maximum approximation error of the Padé approximant is less than the error of the corresponding Taylor polynomial. The advantage of the Padé approximant is more obvious with the increase of the polynomial degree. The bottom row of Table 1 shows the maximum approximation error of the Taylor polynomial is ; however, the maximum approximation error of the Padé approximant is . At the same time, we can find that the form of the Padé approximant is simpler because of its lower degree.

Table 1: Comparison between the Padé approximant and the Taylor polynomial for .

Degree
Polynomial
Maximum approximation error

Padé approximant
Taylor polynomial
Padé
Taylor

3. New Improvements of Jordan’s, Kober’s, and Becker-Stark’s Inequalities

In this section, we give new improvements of Jordan’s, Kober’s, and Becker-Stark’s inequalities based on the Padé approximant.

Theorem 1. For , we have that

Proof. Inequality (13) is equivalent to

It is well known that

By inequality (15), we have that

Then, the proof of inequality (13) is completed.

Theorem 2. For , we have that

Proof. Inequality (17) is equivalent to

It is well known that

By inequality (19), we have that

Then, the proof of inequality (17) is completed.

Theorem 3. For , we have that

where and are defined in inequality (6).

Proof. The left side of inequality (22) is equivalent to

where .
It is obvious that

where , and we have ; then, is a monotony decrease function in (0, ), . Therefore, for .
By the definition of and , we have

, and there exist and such that

which mean inequality (25) is valid [25]. Then, we have . The proof of is obtained from [20].
The proof of Theorem 3 is completed.

4. Conclusions and Analysis

In this paper, a multiple-point Padé approximant method is presented for approximating and bounding some trigonometric functions. We find that the Padé approximant is a better approximation of trigonometric functions. The conclusion is verified in Table 1. We give new refinements and improvements of Jordan’s, Kober’s, and Becker-Stark’s inequalities based on the Padé approximant. In order to compare our results with the previous methods, we introduce the concept of the maximum error. The maximum error is the most important index to measure the upper and lower bounds of an inequality. denotes the maximum error between a function and its lower bound. denotes the maximum error between a function and its upper bound.

Table 2 gives the comparison of the maximum errors between and its bounds for different methods. It is obvious that the results of this paper are superior to the previous conclusions. The upper and lower bounds of inequality (13) are tighter than inequalities (1) and (4). The results of are presented in Table 3. and of inequality (17) is the smallest of three methods in Table 3.

Table 2: Comparison of the maximum errors between and its bounds for different methods.

Method
Error

Bercu [19] (inequality (1))

Zhang et al. [20] (inequality (4))

Results of this paper (inequality (13))

Table 3: Comparison of the maximum errors between and its bounds for different methods.

Method
Error

Bercu [19] (inequality (2))

Zhang et al. [20] (inequality (5))

Results of this paper (inequality (17))

Table 4 gives the comparison of the maximum errors between and its bounds for this paper and Zhang et al.’s paper [20]. Because inequality (3) holds in , not in , we no longer consider the results of literature [19] in the comparison of . Table 4 shows that the upper bound of inequality (6) has a maximum error of ; however, the maximum error of the lower bound of inequality (6) is . This paper greatly improves the lower bound of . The maximum error of the lower bound of inequality (22) in this paper reaches . The maximum error of the lower bound of inequality (22) is far less than that of inequality (6).

Table 4: Comparison of the maximum errors between and its bounds for different methods.

Method
Error

Zhang et al. [20] (inequality (6))

Results of this paper (inequality (22))

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was partially supported by the National Natural Science Foundation of China (nos. 11701152 and 11161038), Key Research Project of Colleges and Universities in Henan Province (no. 20A520016), and Ph.D. Foundation of Henan Polytechnic University (no B2017-44).

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