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

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:


Multiple-Point Padé Approximant Method
e Padé approximant has been studied in many literature studies [19,[21][22][23][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 f(x), let be a rational polynomial interpolating of f(x) at multiple points x 1 , x 2 , . . . , x k such that where E(x) � (1 + q j�1 b j x j ) · f(x) − ( p i�0 a i x i ) and p ≥ 0 and q ≥ 1 are two given integers. ere are p + q + 1 unknowns in equation (9), a i and b j , i � 0, 1, 2, . . . , p, j � 1, 2, . . . , q. By selecting suitable values of l 1 , l 2 , . . . , l k , we can obtain the polynomial R(x) by solving equation (9). e general Padé approximant method is a special case of the multiple-point Padé approximant. Here, we just need to consider one point. If f can be written as a formal power series f(x) � c 0 + c 1 x + c 2 x 2 + · · ·, where the coefficients c j , j � 0, 1, 2, . . ., are constant. Taylor's expansion is one of the most common ways to get a power series of a function.
e Padé approximant R f (x) of degree (p, q) of the function f is determined by e Padé approximant is considered the "best" approximation of a function by a rational function of a given degree. e rational approximation is also good for series with alternation terms and poor polynomial convergence.
is is our motivation of using the Padé approximant to approximate trigonometric functions and improve these trigonometric inequalities. Different values of p and q will affect the approximate performance. By selecting suitable values of p and q, we can obtain the "best" approximant. Let (p, q) � (k, k), and we can obtain a simple result. e result is a special case of (10).
It is well known that where for x ∈ (0, π/2), n ∈ N 0 , and B i 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 c j in formula (10). Let c 2j � 0, c 2j− 1 � T(j), and T(j) is given in (12); we can obtain the Padé approximant of tan(x). 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. e advantage of the Padé approximant is more obvious with the increase of the polynomial degree. e bottom row of Table 1 shows the maximum approximation error of the Taylor polynomial is 6.0401 × 10 − 3 ; however, the maximum approximation error of the Padé approximant is 2.2531 × 10 − 9 . At the same time, we can find that the form of the Padé approximant is simpler because of its lower degree.

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.

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. e 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. e maximum error is the most important index to measure the upper and lower bounds of an inequality. MaxError l denotes the maximum error between a function and its lower bound. MaxError u denotes the maximum error between a function and its upper bound. Table 2 gives the comparison of the maximum errors between sinc(x) and its bounds for different methods. It is obvious that the results of this paper are superior to the previous conclusions. e upper and lower bounds of inequality (13) are tighter than inequalities (1) and (4). e results of cos(x) are presented in Table 3. MaxError l and MaxError u of inequality (17) is the smallest of three methods in Table 3. Table 4 gives the comparison of the maximum errors between tan(x) and its bounds for this paper and Zhang et al.'s paper [20]. Because inequality (3) holds in (0, 1.5701), not in [0, π/2], we no longer consider the Bercu [19] (inequality (1)) 2.5981 × 10 − 3 6.2382 × 10 − 5 Zhang et al. [20] (inequality (4)) 1.0615 × 10 − 6 1.7998 × 10 − 6 Results of this paper (inequality (13)) 1.3042 × 10 − 10 1.3411 × 10 − 8 Table 3: Comparison of the maximum errors between cos(x) and its bounds for different methods.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.