Some New Results of Mitrinovi<i>ć–</i>Cusa’s and Related Inequalities Based on the Interpolation and Approximation Method

Zhang, Lina; Ma, Xuesi

doi:https://doi.org/10.1155/2021/5595650

Journal of Mathematics

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Abstract Introduction Conclusions and Analysis Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5595650 | https://doi.org/10.1155/2021/5595650

Some New Results of Mitrinović–Cusa’s and Related Inequalities Based on the Interpolation and Approximation Method

Lina Zhang¹and Xuesi Ma²

Academic Editor: Dimitri Mugnai

Received25 Feb 2021

Revised22 Mar 2021

Accepted25 Mar 2021

Published15 Apr 2021

Abstract

In this paper, new refinements and improvements of Mitrinović–Cusa’s and related inequalities are presented. First, we give new polynomial bounds for sinc(x) and cos(x) functions using the interpolation and approximation method. Based on the obtained results of the above two functions, we establish new bounds for Mitrinović–Cusa’s, Wilker’s, Huygens’, Wu–Srivastava’s, and Neuman–Sándor’s inequalities. The analysis results show that our bounds are tighter than the previous methods.

1. Introduction

Trigonometric inequalities play an important role in pure and applied mathematics and have been used in various fields. This study starts from the following inequality:which is the focus of many researchers. The left-side inequality of (1) was first proved by Mitrinović [1, 2], which is called as Mitrinović–s inequality. The right-side inequality of (1) was proposed by the German philosopher and theologian Nicolaus de Cusa and proved explicitly by Huygens [3]. Therefore, inequality (1) is called as Mitrinović–Cusa’s inequality.

Huygens’ inequality iswhich is a consequence of inequality (1) [3].

The following inequalitydue to Wilker, is called as Wilker’s inequality [4].

Wu and Srivastava [5] proved the following inequality:

Neuman and Sándor [6] presented the following inequality:

Inequalities (1)–(5) have attracted the attention of many scholars [7–32].

Bhayo and Sándor [33] obtained the following results, for :where and are the best possible constants.

Yang [34] gave the following inequalities:where is the unique root of equation in .

Yang [35] gave further results:where .

Sumner et al. [36] obtained a further result of Wilker’s inequality as follows:

Furthermore, and are the best constants.

Chen and Cheung [37] proved the following inequalities, for :where are the best possible constants.

Nenezic et al. [38] proved the following inequalities, for :

Chen and Paris [39] gave the following inequalities, for :

Wang [40] proved the following inequality:where and are the best possible constants.

Jiang et al. [41] gave the following inequality:where and are the best possible numbers.

In particular, Mortici [42] gave the improvements of inequalities (1)–(5), for every :

Maleevi et al. [43] further improved the above results of inequalities (24)–(29), for and :where , , , , and are Bernoulli’s numbers.

The goal of this paper is to obtain some new inequalities which provide generalizations of inequalities (1)–(5). In order to provide the new bounds of Mitrinović–Cusa’s and related inequalities, we introduce a method called the interpolation and approximation. This method has been successfully applied to prove and approximate a wide category of trigonometric inequalities 26, 27, 44–46.

In this paper, we first give the new polynomial bounds of and functions using the interpolation and approximation method. We obtain the new bounds of Mitrinović–Cusa’s and related inequalities based on the new bounds of the above two functions. The related inequalities include Wilker’s, Huygens’, Wu–Srivastava’s, and Neuman–Sndor’s inequalities. At the same time, we also directly use the interpolation and approximation method to get the upper and lower bounds of Mitrinović–Cusa’s inequality. The analysis results show that our bounds are tighter than the previous conclusions.

2. Main Results

Firstly, we introduce the following theorem of interpolation and approximation which is very useful for our proof [47].

Theorem 1. Let be distinct points in and be integers . Let . Suppose that is a polynomial of degree such that

Then, there exists such that

Next, we try to derive the novel polynomial bounds of and functions using the above interpolation and approximation theorem.

Theorem 2. For , we have thatwhere

Proof. Let and ; then, .
It is easy to see thatWe let ; then,Therefore, is a monotone decreasing function in , and we have ; then, for .
By the definition of and , we haveBy Theorem 1, there exist , such thatwhich mean the conclusions are valid.
The proof of Theorem 2 is completed.

Theorem 3. For , we have thatwhere

Proof. Let and ; then, .
It is obvious that for .
By the definition of and , we haveBy Theorem 1, there exist , such thatwhich mean the conclusions are valid.
The proof of Theorem 3 is completed.
We propose the following refinements and improvements of inequalities (1)–(5).

Theorem 4. For , we have thatwhere , and are defined in Theorems 2 and 3.

Proof. First, we prove inequality (44). By Theorems 2 and 3, we haveTherefore, we have ; then, . Inequality (44) is proved.
Second, we give the proof of inequality (48). For the same reason, we have and ; then, . That is to say, inequality (48) holds.
Third, we prove inequality (49).
It is easy to see that ; then, inequality (49) is valid.
Fourth, we prove inequality (50). By Theorem 2, we have , where , are defined in Theorem 2. Then, can be rewritten aswhere .
Then, , and we have . For the same reason, we can prove . It is obvious that and .
By inequality (55) and the results of , we haveHence, inequality (50) holds.
Fifth, we prove inequality (51). By the result of and , we have . And because of the above results of , we have that inequality (51) holds.
Sixth, we prove inequality (52). By the result of and , we have . And because of the results of and , we have . Therefore, inequality (52) is proved.
At last, we prove inequality (53). By the results of and inequality (54), we have . And because of inequality (55), inequality (53) holds.
The proof of Theorem 4 is completed.
Theorems 2 and 3 give the new bounds of and based on the interpolation and approximation method. Theorem 4 gives the improvements and refinements of Mitrinović–Cusa’s and related inequalities using the new bounds of and . Next, we try to obtain the new improvements of Mitrinović–Cusa’s inequalities directly based on the interpolation and approximation method.

Theorem 5. For , we havewhere

Proof. Let ; then,We let ; then, we have . Then, is a monotone decreasing function in , and we have ; then, for . Therefore, we have .
By the definition of and , we haveBy Theorem 1, there exist , such thatwhich mean the conclusions are valid.

3. Conclusions and Analysis

In this paper, we give the new results of Mitrinović–Cusa’s and related inequalities, including Wilker’s, Huygens’, Wu–Srivastava’s, and Neuman–Sndor’s inequalities. Theorems 2 and 3 present the novel polynomial bounds of and functions using the interpolation and approximation method. Theorem 4 gives the new refinements and improvements of the above five inequalities based on the results of and . 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 (a function). denotes the maximum error between a function and its lower bound. Similarly, denotes the maximum error between a function and its upper bound. Tables 1‒6 give the maximum errors of the upper and lower bounds of the above five inequalities. We consider both sides of Mitrinović–Cusa’s inequality separately, which is why there are six tables for five inequalities. It is obvious that the results of this paper are superior to the previous conclusions.

For Mitrinović’s inequality, many researchers focused on the equivalent form of this inequality, that is, the inequality . Table 1 gives the comparison of the maximum errors between and its bounds for different methods. In inequality (40), the maximum error between the function and the lower bound is only , and the maximum error of the upper bound is . It is easy to see that the maximum errors of this paper are the smallest of all methods. For Cusa’s inequality, Wilker’s inequality, Huygens’ inequality, Wu–Srivastava’s inequality, and Neuman–Sndor’s inequality, the same conclusions can be obtained from Tables 2 to 6.

In this paper, we not only consider the equivalent form of Mitrinović’s inequality but also consider the lower and upper bounds of the function directly. Inequality (44) gives the new improvement of Mitrinović’s inequality. The maximum error between the function and the lower bound is only , and the maximum error of the upper bound is .

Tables 1‒6 show that the results of this paper are far superior to the previous conclusions. Among the previous conclusions, only the conclusions of Maleevi et al. [43] are close to the results of this paper for Cusa’s inequality in Table 2. In this paper, the maximum error between and its lower bound is , and the maximum error between and its upper bound is . The maximum error of the lower bound is , and the maximum error of the upper bound is in Maleevi et al. [43]. Although the maximum errors of Maleevi et al. [43] are close to our results, we can find that the degree of the bounds is 8 in inequality (49), and the degree of the upper bound reaches 10 in inequality (33) (when n = 2).

Another advantage of our results is that the form of the bounds is relatively simple. The bounds are all polynomial functions, and there are no other functions. In previous conclusions, many bounds contain not only polynomial functions but also trigonometric functions, for example, . In general, the polynomial function as the upper and lower bounds of the trigonometric function is more effective than the trigonometric function as the upper and lower bounds of the trigonometric function.

At the end of this paper, we try to obtain the bounds of directly using the interpolation and approximation theorem. Theorem 5 gives the new upper and lower bounds of the function . The last row of Table 2 gives the maximum errors of inequality (56). The maximum error between the function and the lower bound is , and the maximum error of the upper bound is . Table 2 shows that the maximum errors are close to the results of inequality (49). Although the maximum errors of are bigger than the maximum errors of inequality (49), the degree of the bounds is 6 in inequality (56), and the degree of the bounds is 8 in inequality (49). Similarly, we can obtain the upper and lower bounds of other functions directly using the interpolation and approximation theorem.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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

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Copyright © 2021 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.

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