New Refinements and Improvements of Jordan ’ s Inequality

The polynomial bounds of Jordan’s inequality, especially the cubic and quartic polynomial bounds, have been studied and improved in a lot of the literature; however, the linear and quadratic polynomial bounds can not be improved very much. In this paper, new refinements and improvements of Jordan’s inequality are given. We present new lower bounds and upper bounds for strengthened Jordan’s inequality using polynomials of degrees 1 and 2. Our bounds are tighter than the previous results of polynomials of degrees 1 and 2. More importantly, we give new improvements of Jordan’s inequality using polynomials of degree 5, which can achieve much tighter bounds than those previous methods.


Introduction
The following inequality with equality holds if and only if x = π/2, is the famous Jordan's inequality [1].sinc(x) is also called the "sampling function" that arises frequently in signal processing and the theory of Fourier transforms.The Jordan's inequality plays an important role in many areas of pure and applied mathematics.Many improvements and refinements of Jordan's inequality were presented in the recent period [2][3][4][5][6][7].There are some sharp lower and upper bounds for the sinc(x) function by using polynomial degrees from 1 to 4. Zhang et al. [8] gave that Qi et al. [9] proved that 2 π Deng [10] presented and Jiang et al. [11], similarly, gave the result Equalities in Labels ( 2)-( 5) are valid if and only if x = π/2.As x → 0 + , the equalities on the right-hand sides of (3)-( 5) are valid, but strict inequalities on the left-hand sides of (3)-( 5) and two sides of (2) persist.Debnath et al. [12] gave the improvements of (3) and ( 5) and where equalities on two sides of ( 6) and ( 7) are valid; however, as x → π 2 − , the lower and upper limits of ( 6) and ( 7) are different from that of sinc(x).The problem of strict inequalities of ( 6) and ( 7) still exists.
In order to ensure that the equality of Jorand's inequality is valid near zero and π/2, Agarwal et al. [13] and Chen et al. [14] gave new lower and upper bounds by using polynomials of degree of 3 and 4, where g l 3,A (x) = 1 + 4(66−43π+7π 2 ) π 2 x − 4(124−83π+14π 2 ) π 3 x 2 − 4(12−4π) x 2 + 16(π−3) x 4 .Zeng and Wu [15] gave the polynomial bounds of degree m(m ≥ 2) for sinc(x) Putting m = 2, 3, 4 in (11) results in (3), ( 4) and ( 5), respectively.The cubic and quartic polynomial lower and upper bounds of sinc(x) have been improved in a lot of literature; however, the linear and quadratic polynomial lower and upper bounds can not be improved very well.To give new tighter linear and quadratic polynomial bounds is the first aim of the paper.The second aim is to further refine and generalize the Jordan's inequality.
The paper gives improvements of the polynomial bounds of degrees 1 and 2.More importantly, we present new improvements of Jordan's inequality using polynomials of degree 5, which can achieve much tighter bounds than those previous methods.

Results
In this section, we will give some results about the n-th-order derivative and two-sides bounds of sinc(x).Firstly, we present a Lemma that is very useful for our proof [16].
Then, there exists Next, we give Theorems of n-th-order derivative and two sides bounds of sinc(x) using polynomials of degrees 1 and 2.
Proof.For the definition of sinc(x), we have ).
The proof of Theorem 1 is completed.
Theorems 2 and 3 give new bounds of sinc(x) using polynomials of degrees 1 and 2. Figures 1 and 2 give the error between sinc(x) and the polynomial bounds from degree 1 to 2. Both figures show that our bounds are tighter than the previous results.The same conclusion can also be shown in Table 1.Error low and Error upp denote the maximum errors between sinc(x) and the lower and upper bounds, respectively.It is obvious that the maximum errors are less than or equal to those of previous methods using polynomials of degrees 1 and 2.
Figure 5 gives the errors between sinc(x) and the polynomial bounds of degrees 4 and 5.For the same reason, we also give the errors between sinc(x) and the polynomial bounds of Equations ( 10) and ( 14) in Figure 6.
Figure 7 gives a comparison between Equations ( 14) and ( 11); here, we set m = 5, 8, 10 in Equation (11), where m is the degree of the polynomial.Our results are obviously better than that of Zeng [15]; meanwhile, we find that the error is even greater with the increase of m's value of Equation (11).Maximum errors of different methods are presented in Table 1.Although Equation (11) gives the polynomial bounds of degree m for sinc(x), the error of Equation ( 11) is relatively large.

Conclusions
In this paper, we gave new refinements and improvements of Jordan's inequality.Firstly, the new polynomial bounds of degrees 1 and 2 were given.The results show that our bounds are tighter than the previous results of polynomials of degrees 1 and 2. Meanwhile, we presented new improvements of Jordan's inequality using polynomials of degree 5, which can achieve much tighter bounds than those previous methods.
Much work still remains.The polynomial bounds of degree 5 were given in this paper, and the polynomial bounds of higher degree are needed for tighter bounds.However, it will require more complicated calculations.Furthermore, it is still an important problem to find tighter polynomial bounds of lower degrees.
Author Contributions: All authors contributed equally in writing this article.All authors read and approved the final manuscript.

Funding:
The work is partially supported by the National Natural Science Foundation of China (No. 11701152, 11161038).

Figure 3 .
Figure 3. Error plots between sinc(x) and the polynomial bounds of degrees 3 and 5.

Figure 5 .
Figure 5. Error plots between sinc(x) and the polynomial bounds of degrees 4 and 5.

Table 1 .
Maximum errors from different methods.