Journal of Mathematics

Journal of Mathematics / 2021 / Article

Research Article | Open Access

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

Lina Zhang, Xuesi Ma, "Some New Results of Mitrinović–Cusa’s and Related Inequalities Based on the Interpolation and Approximation Method", Journal of Mathematics, vol. 2021, Article ID 5595650, 13 pages, 2021. https://doi.org/10.1155/2021/5595650

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

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 [732].

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, 4446.

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