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

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. (e analysis results show that our bounds are tighter than the previous methods.


Introduction
Trigonometric inequalities play an important role in pure and applied mathematics and have been used in various fields. is study starts from the following inequality: (cos(x)) ( which is the focus of many researchers. e left-side inequality of (1) was first proved by Mitrinović [1,2], which is called as Mitrinović-s inequality. e right-side inequality of (1) was proposed by the German philosopher and theologian Nicolaus de Cusa and proved explicitly by Huygens [3]. erefore, inequality (1) is called as Mitrinović-Cusa's inequality.
is 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 sin c(x) and cos(x) 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. e related inequalities include Wilker's, Huygens', Wu-Srivastava's, and Neuman-Sa � ndor'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. e analysis results show that our bounds are tighter than the previous conclusions.

Main Results
Firstly, we introduce the following theorem of interpolation and approximation which is very useful for our proof [47].
and n 0 , n 1 , · · · n r be r + 1 integers ≥0. Let N � n 0 + · · · + n r + r. Suppose that g(t) is a polynomial of degree N such that Journal of Mathematics Next, we try to derive the novel polynomial bounds of sin c(x) and cos(x) functions using the above interpolation and approximation theorem. where

Journal of Mathematics
Proof.

Journal of Mathematics
where s l (x), s u (x), c l (x), and c u (x) are defined in eorems 2 and 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-Sa � ndor's inequalities. eorems 2 and 3 present the novel polynomial bounds of sin c(x) and cos(x) functions using the interpolation and approximation method. eorem 4 gives the new refinements and improvements of the above five inequalities based on the results of sin c(x) and cos(x). In order to compare our results with the previous methods, we introduce the concept of the maximum error. e maximum error is the most important index to measure the upper and lower bounds of an inequality (a function). MaxError low denotes the maximum error between a function and its lower bound. Similarly, MaxError low 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 (sin(x)/x) 3 − cos(x) > 0. Table 1 gives the comparison of the maximum errors between (sin(x)/x) 3 − cos(x) > 0 and its bounds for different methods. In inequality (40), the maximum error between the function (sin(x)/x) 3 − cos(x) > 0 and the lower bound is only 3.2409 × 10 − 8 , and the maximum error of the upper bound is 3.1707 × 10 − 8 . 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-Sa � ndor'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 (sin(x)/x) − cos (x) (1/3) directly. Inequality (44) gives the new improvement of Mitrinović's inequality. e maximum error between the function (sin(x)/x) − cos (x) (1/3) and the lower bound is only 1.0834 × 10 − 8 , and the maximum error of the upper bound is 6.8129 × 10 − 8 .
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 Male� sevic � 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 (sin(x)/x) − (2 + cos(x)/3) and its lower bound is 1.0803 × 10 − 8 , and the maximum error between (sin(x)/x) − (2 + cos(x)/3) and its upper bound is 1.0701 × 10 − 8 . e maximum error of the lower bound is 5.9603 × 10 − 6 , and the maximum error of the upper bound is 1.1839 × 10 − 7 in Male� sevic � et al. [43]. Although the maximum errors of Male� sevic � 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. e bounds are all polynomial Table 5: Maximum errors between (x/sin(x)) 2 + (x/tan(x)) and its bounds (Wu and Srivastava's inequality).
At the end of this paper, we try to obtain the bounds of (sin(x)/x) − (2 + cos(x)/3) directly using the interpolation and approximation theorem. eorem 5 gives the new upper and lower bounds of the function (sin(x)/x) − (2 + cos(x)/3). e last row of Table 2 gives the maximum errors of inequality (56). e maximum error between the function (sin(x)/x) − (2 + cos(x)/3) and the lower bound is 5.8244 × 10 − 7 , and the maximum error of the upper bound is 3.2174 × 10 − 6 . 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
e authors declare that they have no conflicts of interest.