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

Authors: Lina Zhang; Xuesi Ma
Journal: Journal of Mathematics
 (2020)

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## Abstract

A multiple-point Padé approximant method is presented for approximating and bounding some trigonometric functions in this paper. We give new refinements and improvements of some trigonometric inequalities including Jordan’s inequality, Kober’s inequality, and Becker-Stark’s inequality. The analysis results show that our conclusions are better than the previous conclusions.


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## Body

## 1. Introduction

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:(1)−7x2+603x2+60<sinxx<11x4−360x2+252060x2+2520,0<x<π2,(2)17x4−480x2+10802x4+60x2+1080<cosx<3x4−56x2+1204x2+120,0<x<π2,(3)−28x4−600x2+72009x6+12x4−3000x2+7200<tanxx<22x8−60x6−4680x4−237600x2+27216001020x6+14040x4−1144800x2+2721600,0<x<1.5701.Zhang et al. [20] gave the improvements of inequalities (1)–(3):(4)−5x6+364x4−9240x2+60480840x2+72≤sinxx≤551x4−22260x2+166320155x4+364x2+11088,0≤x≤π2,(5)−13x6+660x4−9720x2+20160360x2+56<cosx<313x4−6900x2+1512013x4+660x2+15120,0≤x≤π2,(6)21495−60x2+x410395−4725x2+210x4−x6<tanxx<T1x105π2−4x2⋅T2x,0≤x≤π2,where(7)T1x=π6−840π4+75600π2−665280x6+210π6+52920π4−7620480π2+69854400x4+−17955π6+1323000π4+52390800π2−628689600x2+155925π4−112π2+1008π2,T2x=26π4−2664π2+23670x4+−666π4+73980π2−665280x2+1485π4−166320π2+1496880.In this paper, we present a multiple-point Padé approximant method for approximating and bounding some trigonometric functions. New refinements and improvements of related trigonometric inequalities are obtained, including Jordan’s inequality, Kober’s inequality, and Becker-Stark’s inequality. We introduce the concept of maximum error to compare our results with the previous methods. The results show that our bounds are tighter than the previous conclusions.


## 2. Multiple-Point Padé Approximant Method

The Padé approximant has been studied in many literature studies [19, 21–24]. 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 fx, let(8)Rx=∑i=0paixi1+∑j=1qbjxj,be a rational polynomial interpolating of fx at multiple points x1,x2,…,xk such that(9)Ei1x1=0,Ei2x2=0,…,Eikxk=0,i1=0,1,…,l1,i2=0,1,…,l2,…,ik=0,1,…,lk,where Ex=1+∑j=1qbjxj⋅fx−∑i=0paixi and p≥0 and q≥1 are two given integers. There are p+q+1 unknowns in equation (9), ai and bj, i=0,1,2,…,p,j=1,2,…,q. By selecting suitable values of l1,l2,…,lk, we can obtain the polynomial Rx by solving equation (9).The general Padé approximant method is a special case of the multiple-point Padé approximant. Here, we just need to consider one point. Iff can be written as a formal power series fx=c0+c1x+c2x2+⋯, where the coefficients cj,j=0,1,2,…, are constant. Taylor’s expansion is one of the most common ways to get a power series of a function. The Padé approximant Rfx of degree p,q of the function f is determined by(10)a0=c0,a1=c0b1+c1,a2=c0b2+c1b1+c2,⋅⋅⋅ap=c0bp+⋯+cp−1b1+cp,0=cp+1+cpb1+⋯+cp−q+1bq,⋅⋅⋅0=cp+q+cp+q−1b1+⋯+cpbq.The Padé approximant is considered the “best” approximation of a function by a rational function of a given degree. The rational approximation is also good for series with alternation terms and poor polynomial convergence. This is our motivation of using the Padé approximant to approximate trigonometric functions and improve these trigonometric inequalities. Different values ofp 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. The result is a special case of (10).It is well known that(11)tanx=∑n=1∞Tnx2n−1,where(12)Tn=22n22n−12n!B2n,for x∈0,π/2,n∈N0, and Bi 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 cj in formula (10). Let c2j=0,c2j−1=Tj, and Tj is given in (12); we can obtain the Padé approximant of tanx. In the same way, we can also obtain the Padé approximants of other trigonometric functions.Table1 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. The advantage of the Padé approximant is more obvious with the increase of the polynomial degree. The 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.Table 1
Comparison between the Padé approximant and the Taylor polynomial fortanx.
DegreePolynomialMaximum approximation errorPadé approximantTaylor polynomialPadéTaylor1,1/2xx5.5741×10−15.5741×10−12,2/43x/3−x2x+1/3x35.7408×10−22.2407×10−13,3/615x−x3/15−6x2x+1/3x3+2/15x51.8522×10−39.0741×10−24,4/8105x−10x3/105−45x2+x4x+1/3x3+2/15x5+17/315x73.0675×10−53.6773×10−25,5/10945x−105x3+x5/945−420x2+15x4x+1/3x3+2/15x5+17/315x7+62/2835x93.1725×10−71.4903×10−26,6/1210395x−1260x3+21x5/10395−4725x2+210x4−x6x+1/3x3+2/15x5+17/315x7+62/2835x9+1382/155925x112.2531×10−96.0401×10−3


## 3. 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.Theorem 1.
For∀x∈Γ=0,π/2, we have that(13)3113510400−498960000x2+22619520x4−451440x6+4620x8−23x1019958400156+x2≤sinxx≤19958400−3144960x2+136080x4−2448x6+19x8181440110+x2.Proof.
Inequality (13) is equivalent to(14)19958400156+x2sinx−3113510400−498960000x2+22619520x4−451440x6+4620x8−23x10x≥0,181440110+x2sinx−19958400−3144960x2+136080x4−2448x6+19x8x≤0.
It is well known that(15)αx−x151307674368000≤sinx≤x−x36+x5120−x75040+x9362880−x1139916800+x136227020800=αx.
By inequality (15), we have that(16)19958400156+x2sinx−3113510400−498960000x2+22619520x4−451440x6+4620x8−23x10x≥19958400156+x2αx−x151307674368000−3113510400−498960000x2+22619520x4−451440x6+4620x8−23x10x=17x151135134054−x2≥0,∀x∈Γ,181440110+x2sinx−19958400−3144960x2+136080x4−2448x6+19x8x≤181440110+x2αx−19958400−3144960x2+136080x4−2448x6+19x8x=x1334320x2−46≤0,∀x∈Γ.
Then, the proof of inequality (13) is completed.Theorem 2.
For∀x∈Γ, we have that(17)131040−62160x2+3814x4−59x6265520+1680x2+17x4≤cosx≤1814400−887040x2+65520x4−1680x6+17x82016090+x2.Proof.
Inequality (17) is equivalent to(18)265520+1680x2+17x4cosx−131040−62160x2+3814x4−59x6≥0,2016090+x2cosx−1814400−887040x2+65520x4−1680x6+17x8≤0.
It is well known that(19)βx−x1487178291200≤cosx≤1−x22+x424−x6720+x840320−x103628800+x12479001600=βx.
By inequality (19), we have that(20)265520+1680x2+17x4cosx−131040−62160x2+3814x4−59x6≥265520+1680x2+17x4βx−x1487178291200−131040−62160x2+3814x4−59x6=x12435891456008321040−168168x2+1414x4−17x6≥x12435891456008321040−168168×22−17×26≥0,∀x∈Γ,(21)2016090+x2cosx−1814400−887040x2+65520x4−1680x6+17x8≤2016090+x2βx−1814400−887040x2+65520x4−1680x6+17x8=x1223760−42+x2≤0,∀x∈Γ.
Then, the proof of inequality (17) is completed.Theorem 3.
For∀x∈Γ, we have that(22)T3xT4x<tanxx<T1x105π2−4x2⋅T2x,0≤x≤π2,where T1x and T2x are defined in inequality (6).(23)T3x=−16796163−26542082−710208π+163762563+212336642−35230464x8+π30792963+46448642+1270080π−277136643−398131202+62923392x7+π2−22978083−32808962−934596π+188694363+302284802−45416304x6+π38942403+11980802+366210π−66324423−118702082+16935408x5+π4−1914843−2385922−81833π+12721053+25354242−3429916x4+π5213843+245762+202952π−25296332−2785282+355544x3+π6−9723−10242−11532π+51033+122882−14572x2+π8,(24)T4x=π−2799363−6635522−150336π+50388483+26542082−7537536x7+π26065283+14929922+347328π−116173443−53084162+16604352x6+π3−5462643−14008322−337140π+112091043+42762242−15206904x5+π42650323+7096322+178146π−58601883−17694722+7497936x4+π5−748443−2094082−55319π+17904243+3952642−2148668x3+π6123123+359682+201792π−3190593−450562+357852x2+π7−218732−33282−1000π+306183+20482−641112x1+π88132+1282+832π−12153+1185+π9.Proof.
The left sideT3x/T4x<tanx/x of inequality (22) is equivalent to(25)e3x=sincx−T3x≥0,e4x=cosx−T4x≤0,where sincx=sinx/x.
It is obvious that(26)sinc9x=gxx10,(27)cos9x=−sinx≤0,where gx=x9−72x7+3024x5−60480x3+362880xcosx+−9x8+504x6−1520x4+181440x2−362880sinx, and we have g′x=−x9sinx≤0; then, gx is a monotony decrease function in (0, π/2), gx≥g0=0. Therefore, sinc9x≤0 for x∈0,π/2.
By the definition ofe3x and e4x, we have(28)e30=e3π6=e3π4=e3π2=e3′0=e3′π6=e3′π4=e3′π3=0,(29)e40=e4π6=e4π4=e4π2=e4′π6=e4′π4=e4′π3=e4′π2=0,∀x∈Γ, and there exist ξ∈0,π/2 and η∈0,π/2 such that(30)e3x=sinc9ξ9!x2x−π62x−π42x−π32x−π2≥0,(31)e4x=cos9η9!x2x−π62x−π42x−π32x−π22≤0,which mean inequality (25) is valid [25]. Then, we have T3x/T4x<tanx/x. The proof of tanx/x<T1x/105π2−4x2⋅T2x is obtained from [20].
The proof of Theorem3 is completed.


## 4. 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. The conclusion is verified in Table1. 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. The maximum error is the most important index to measure the upper and lower bounds of an inequality. MaxErrorl denotes the maximum error between a function and its lower bound. MaxErroru denotes the maximum error between a function and its upper bound.Table2 gives the comparison of the maximum errors between sincx and its bounds for different methods. It is obvious that the results of this paper are superior to the previous conclusions. The upper and lower bounds of inequality (13) are tighter than inequalities (1) and (4). The results of cosx are presented in Table 3. MaxErrorl and MaxErroru of inequality (17) is the smallest of three methods in Table 3.Table 2
Comparison of the maximum errors betweensincx and its bounds for different methods.
MethodErrorMaxErrorlMaxErroruBercu [19] (inequality (1))2.5981×10−36.2382×10−5Zhang et al. [20] (inequality (4))1.0615×10−61.7998×10−6Results of this paper (inequality (13))1.3042×10−101.3411×10−8Table 3
Comparison of the maximum errors betweencosx and its bounds for different methods.
MethodErrorMaxErrorlMaxErroruBercu [19] (inequality (2))6.8710×10−46.8832×10−4Zhang et al. [20] (inequality (5))1.3987×10−52.9435×10−5Results of this paper (inequality (17))2.9194×10−72.0648×10−7Table4 gives the comparison of the maximum errors between tanx 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 results of literature [19] in the comparison of tanx. Table 4 shows that the upper bound of inequality (6) has a maximum error of 2.7941×10−6; however, the maximum error of the lower bound of inequality (6) is 2.0690×10−1. This paper greatly improves the lower bound of tanx. The maximum error of the lower bound of inequality (22) in this paper reaches 3.0498×10−5. The maximum error of the lower bound of inequality (22) is far less than that of inequality (6).Table 4
Comparison of the maximum errors betweentanx and its bounds for different methods.
MethodErrorMaxErrorlMaxErroruZhang et al. [20] (inequality (6))2.0690×10−12.7941×10−6Results of this paper (inequality (22))3.0498×10−52.7941×10−6



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*Source: 2753691-2020-05-19.xml*