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

Authors: Lina Zhang; Xuesi Ma
Journal: Journal of Mathematics
 (2021)

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## 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.


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## Body

## 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:(1)cosx1/3<sincx=sinxx<2+cosx3,0<x<π2,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 is(2)2sinxx+tanxx>3,0<x<π2,which is a consequence of inequality (1) [3].The following inequality(3)sinxx2+tanxx>2,0<x<π2,due to Wilker, is called as Wilker’s inequality [4].Wu and Srivastava [5] proved the following inequality:(4)xsinx2+xtanx>2,0<x<π2.Neuman and Sándor [6] presented the following inequality:(5)3xsinx+cosx>4,0<x<π2.Inequalities (1)–(5) have attracted the attention of many scholars [7–32].Bhayo and Sándor [33] obtained the following results, for x∈0,π/2:(6)2+cosx−x/π33<sinxx<2+cosx−x/π43,(7)sinxx<π2−x2π2+αx2<2+cosx3<π2−x2π2+βx2,where α=π2/6−1 and β=1/2 are the best possible constants.Yang [34] gave the following inequalities:(8)sinxx>cospx1/3p2,0<x<π2,p∈p1,1,sinxx<cosqx1/3q2,0<x<π2,q∈0,5/5,where p1=0.45346830977067… is the unique root of equation ln2/π−1/3p2lncospπ/2=0 in p∈0,1.Yang [35] gave further results:(9)2p+p+3cosx3p+1+2cosx<cosx1/3<2q+q+3cosx3q+1+2cosx<sinxx<2r+r+3cosx3r+1+2cosx<2+cosx3<2s+s+3cosx3s+1+2cosx,where x∈0,π/2,p=0,q∈1,π−3−1,r∈9,∞,ands∈−∞,−1.Sumner et al. [36] obtained a further result of Wilker’s inequality as follows:(10)2+16π4x3tanx<sinxx2+tanxx<2+845x3tanx,0<x<π2.Furthermore,16/π4 and 8/45 are the best constants.Chen and Cheung [37] proved the following inequalities, for x∈0,π/2:(11)3+320x3tanx<2sinxx+tanxx<3+2π4x3tanx,(12)3+320x4+356x5tanx<2sinxx+tanxx<3+320x4+2π6x5tanx,(13)2+845x4+16315x5tanx<sinxx2+tanxx<2+845x4+2π6x5tanx,(14)2+845x4+16315x6+1044725x7tanx<sinxx2+tanxx<2+845x4+16315x6+2π8x7tanx,(15)xsinx2+xtanx<2+245x3tanx,(16)xsinx2+xtanx<2+245x4+8945x5tanx,where 3/20,2/π4,3/56,2/π6,16/315,104/4725,2/π8,2/45,and8/945 are the best possible constants.Nenezic et al. [38] proved the following inequalities, for x∈0,π/2:(17)2+845−8945x2x3tanx<sinxx2+tanxx<2+845−8945x2+480π6−40320π4+362880014175π8x4x3tanx,(18)2+16π4+160π5−16π3π2−xx3tanx<sinxx2+tanxx<2+16π4+160π5−16π3π2−x+960π6−96π4π2−x2x3tanx.Chen and Paris [39] gave the following inequalities, for x∈0,π/2:(19)3+320+1280x2+2333600x4x3tanx<2sinxx+tanxx<3+320+1280x2+17920−168π4−π670π8x4x3tanx,(20)2+845−8945x2+1614175x4x3tanx<sinxx2+tanxx<2+845−8945x2+241920−2688π4+32π6945π8x4x3tanx,(21)2+245−2315x2−224−8π2315π4x4x3tanx<xsinx2+xtanx<2+245−2315x2−41575x4x3tanx.Wang [40] proved the following inequality:(22)2+245x3sinx<xsinx2+xtanx<2+2π−16π3x3sinx,0<x<π2,where 2/45 and 2/π−16/π3 are the best possible constants.Jiang et al. [41] gave the following inequality:(23)4+110x3sinx<3xsinx+cosx<4+12π−32π3x3sinx,0<x<π2,where 1/10 and 12π−32/π3 are the best possible numbers.In particular, Mortici [42] gave the improvements of inequalities (1)–(5), for every 0<x<π/2:(24)−x415<cosx−sinxx3<−x415+23x61890,(25)−1180x4<sinxx−cosx+23<−1180x4+13780x6,(26)3+320x4−3140x61cosx<2sinxx+tanxx<3+320x4cosx,(27)2+845x4−8105x61cosx<sinxx2+tanxx<2+845x4cosx,(28)xsinx2+xtanx>2+245x4,(29)3xsinx+cosx>4+110x4+1210x6.Malesˇevic´ et al. [43] further improved the above results of inequalities (24)–(29), for x∈0,π/2 and n∈N:(30)∑k=22n−1kAxx2k<cosx−sinxx3<∑k=22n+1−1kAkx2k,(31)∑k=22n−1k+1Bxx2k<sinxx−cosx+23<∑k=22n+1−1k+1Bkx2k,(32)3+1cosx∑k=22n+1−1kCkx2k<2sinxx+tanxx<3+1cosx∑k=22n−1kCkx2k,(33)2+1cosx∑k=22n+1−1kDxx2k<sinxx2+tanxx<2+1cosx∑k=22n−1kDkx2k,(34)2+∑k=2m−1B2k2k−24k2k!x2k+2xπ2mπ24−2−∑k=2m−1B2k2k−24k2k!π22k>xsinx2+xtanx>2+∑k=2mB2k2k−24k2k!x2k,(35)4+∑k=1m−13B2k22k−2+−1k2k!x2k+2xπ2m3π2−4−∑k=1m−13B2k22k−2+−1k2k!π22k>3xsinx+cosx>4+∑k=1m3B2k22k−2+−1k2k!x2k,where Ak=32k+3−32k3−96k2−88k−27/42k+3!, Bk=2/3k−1/2k+1!, Ck=24k−3k−1/2k+1!, Dk=1/4−9+32k+2−40k−32k2/2k+2!, and Bi 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, 44–46.In this paper, we first give the new polynomial bounds ofsincx and cosx 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–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. 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.
Letw0,w1,…,wr be r+1 distinct points in a,b and n0,n1,⋯nr be r+1 integers ≥0. Let N=n0+⋯+nr+r. Suppose that gt is a polynomial of degree N such that(36)giwj=fiwj,i=0,…,nj,j=0,…,r.Then, there existsξt∈a,b such that(37)ft−gt=fN+1ξtN+1!∏i=0rt−wini+1.Next, we try to derive the novel polynomial bounds ofsincx and cosx functions using the above interpolation and approximation theorem.Theorem 2.
Forx∈0,π/2, we have that(38)slx=1+b2x2+b3x3+b4x4+b5x5+b6x6+b7x7+b8x8≤sincx≤1+a1x+a2x2+a3x3+a4x4+a5x5+a6x6+a7x7+a8x8=sux,where(39)a1=1π2−5768+20482+801932−30322+5122+2433π,a2=1π3146932−450562−1071633+7331/2+133122+58323π,a3=1π4−1542128+3952642+11926443−39696+1438722+585633π,a4=1π58694420−17694722−71558643+236620+8376322+3193023π,a5=1π6−28539504+42762242+249784563−828216+28385282+10215723π,a6=1π754670032−53084162−507034083+1694232+56033282+19187283π,a7=1π8−56681856+26542082+554273283−1871424+59719682+19595523π,a8=1π924572160−251942403+860544+26542082+8398083π,b2=1π3−14572+122882+51033−11532+10242+9723π,b3=1π4355544−2785282−2529633/2+202952+245762+213843π,b4=1π5−3429916+25354242+12721053−81833+2385922+1914843π,b5=1π616935408−118702082−66324423+366210+11980802+8942403π,b6=1π7−45416304+302284802+188694363−934596+32808962+22978083π,b7=1π862923392−398131202−277136643+1270080+46448642+30792963π,b8=1π9−35230464+212336642+163762563−710208+26542082+16796163π.Proof.
Leteslx=sincx−1−b2x2−b3x3−b4x4−b5x5−b6x6−b7x7−b8x8 and esux=sincx−1−a1x−a2x2−a3x3−a4x4−a5x5−a6x6−a7x7−a8x8; then, esl9x=esu9x=sinc9x.
It is easy to see that(40)sinc9x=−9x8+504x6−15120x4+181440x2−362880sinx+x9−72x7+3024x5−60480x3+362880xcosxx10.
We letgx=−9x8+504x6−15120x4+181440x2−362880sinx+x9−72x7+3024x5−60480x3+362880xcosx; then,(41)g′x=−x9sinx<0,0<x<π2.
Therefore,gx is a monotone decreasing function in 0,π/2, and we have gx≤g0=0; then, sinc9x≤0 for x∈0,π/2.
By the definition ofeslx and esux, we have(42)esu0=esuπ6=esuπ4=esuπ3=esuπ2=esu′π6=esu′π4=esu′π3=esu′π2=0,esl0=eslπ6=eslπ4=eslπ3=eslπ2=esl′0=esl′π6=esl′π4=esl′π3=0.
By Theorem1, there exist ζj,j=1,2, such that(43)esux=sinc9ζ19!xx−π62x−π42x−π32x−π22≤0,eslx=sinc9ζ29!x2x−π62x−π42x−π32x−π2≥0,which mean the conclusions are valid.
The proof of Theorem2 is completed.Theorem 3.
Forx∈0,π/2, we have that(44)clx=1+d2x2+d3x3+d4x4+d5x5+d6x6+d7x7+d8x8≤cosx≤1+c1x+c2x2+c3x3+c4x4+c5x5+c6x6+c7x7+c8x8=cux,where(45)c1=1π1185−12153+832+1282+8132π,c2=1π2−641112+20482+306183−1000+33282+218732π,c3=1π3357852−450562−3190593+201792+359682+123123π,c4=1π4−2148668+3952642+17904243−55319+2094082+748443π,c5=1π57497936−17694722−58601883+178146+7096322+2650323π,c6=1π6−15206904+42762242+112091043−337140+14008322+5462643π,c7=1π716604352−53084162−116173443+347328+14929922+6065283π,c8=1π8−7537536+26542082+50388483−150336+6635522+2799363π,d2=1π222492+20482−34023+162+2562+8132π,d3=1π3−32013−450562+806763−3564+61442+202532π,d4=1π4342202+3952642−7635063+31914+596482+102873π,d5=1π5−1844604−17694722+37188723−149040+2995202+542703π,d6=1π65355216+42762242−98735763+382968+8202242+1564923π,d7=1π7−7967808−53084162+135768963−513216+11612162+2332803π,d8=1π84748544+26542082−75582723+279936+6635522+1399683π.Proof.
Leteclx=cosx−1−d2x2−d3x3−d4x4−d5x5−d6x6−d7x7−d8x8 and ecux=cosx−1−c1x−c2x2−c3x3−c4x4−c5x5−c6x6−c7x7−c8x8; then, ecl9x=ecu9x=cos9x.
It is obvious thatcos9x=−sinx≤0 for x∈0,π/2.
By the definition ofeclx and ecux, we have(46)ecu0=ecuπ6=ecuπ4=ecuπ3=ecuπ2=ecu′π6=ecu′π4=ecu′π3=ecu′π2=0,ecl0=eclπ6=eclπ4=eclπ3=eclπ2=ecl′0=ecl′π6=ecl′π4=ecl′π3=0.
By Theorem1, there exist ηj,j=1,2, such that(47)ecux=cos9η19!xx−π62x−π42x−π32x−π22≤0,eclx=cos9η29!x2x−π62x−π42x−π32x−π2≥0,which mean the conclusions are valid.
The proof of Theorem3 is completed. □
We propose the following refinements and improvements of inequalities (1)–(5).Theorem 4.
Forx∈0,π/2, we have that(48)slx−cux1/3≤sinxx−cosx1/3≤sux−clx1/3,(49)slx3−cux≤sinxx3−cosx≤sux3−clx,(50)3slx−cux−23≤sinxx−2+cosx3≤3sux−clx−23,(51)2slx+slxcux≤2sinxx+tanxx≤2sux+suxclx,(52)slx2+slxcux≤sinxx2+tanxx≤sux2+suxclx,(53)1sux2+clxsux≤xsinx2+xtanx≤1slx2+cuxslx,(54)3sux+clx≤3xsinx+cosx≤3slx+cux,where slx,sux,clx, and cux are defined in Theorems 2 and 3.Proof.
First, we prove inequality (44). By Theorems 2 and 3, we have(55)slx≤sinxx≤sux,(56)clx≤cosx≤cux.
Therefore, we haveclx1/3≤cosx1/3≤cux1/3; then, slx−cux1/3≤sinx/x−cosx1/3≤sux−clx1/3. Inequality (44) is proved.
Second, we give the proof of inequality (48). For the same reason, we have slx3<sinx/x3<sux3 and −cux≤−cosx≤−clx; then, slx3−cux≤sinx/x3−cosx≤sux3−clx. That is to say, inequality (48) holds.
Third, we prove inequality (49).
It is easy to see that2+clx/3≤2+cosx/3≤2+cux/3; then, inequality (49) is valid.
Fourth, we prove inequality (50). By Theorem 2, we have clx=1+d2x2+d3x3+d4x4+d5x5+d6x6+d7x7+d8x8, where di,i=2,3,…,8, are defined in Theorem 2. Then, clx can be rewritten as(57)clx=x−x1x−x2x−x3x−x4x−x5x−x6x−x7x−x8,where x1≈−7.01106143,x2≈5.18438798+2.25095656i,x3≈5.18438798−2.25095656i,x4≈4.42109817,x5≈−4.444313683+1.43538013i,x6≈−4.444313683−1.43538013i,x7≈1.57079633,x8≈−1.56923028.
Then,x∈0,π/2, and we have clx>0. For the same reason, we can prove slx>0,x∈0,π/2. It is obvious that cux>0 and sux>0.
By inequality (55) and the results of clx>0,slx>0,cux>0,andsux>0, we have(58)2slx≤2sinxx≤2sux,slxcux≤tanxx≤suxclx.
Hence, inequality (50) holds.
Fifth, we prove inequality (51). By the result of slx>0 and sux>0, we have slx2<sinx/x2<sux2. And because of the above results of slx/cux≤tanx/x≤sux/clx, we have that inequality (51) holds.
Sixth, we prove inequality (52). By the result of slx>0 and sux>0, we have slx2<sinx/x2<sux2. And because of the results of slx/cux≤tanx/x≤sux/clx and slx>0,sux>0,clx>0,cux>0, we have clx/sux<x/tanx<cux/slx. Therefore, inequality (52) is proved.
At last, we prove inequality (53). By the results of slx>0,sux>0 and inequality (54), we have 1/sux<x/sinx<1/slx. And because of inequality (55), inequality (53) holds.
The proof of Theorem4 is completed. □
Theorems2 and 3 give the new bounds of sincx and cosx based on the interpolation and approximation method. Theorem 4 gives the improvements and refinements of Mitrinović–Cusa’s and related inequalities using the new bounds of sincx and cosx. Next, we try to obtain the new improvements of Mitrinović–Cusa’s inequalities directly based on the interpolation and approximation method.Theorem 5.
Forx∈0,π/2, we have(59)g2x2+g3x3+g4x4+g5x5+g6x6≤sinxx−2+cosx3≤f1x+f2x2+f3x3+f4x4+f5x5+f6x6,where(60)f1=1π2208−25602+510332−7532−1282π−43+3223+2732π2,f2=1π3−3296+378882−371793+163043−588823π+64/3+44823+40532π2,f3=1π420560−2176002+2114103−30962−3468823π−4043+246423+11883π2,f4=1π5−62976+6082562−5861163+2588243−988162/3π+12563+665623+34023π2,f5=1π694464−8294402+7931523−117472−455682π−640+29442+47523π2,f6=1π7−55296+4423682−4199043+62592−245762π+384+15362+25923π2,g2=1π332−30722+36453−17683−25623π−6423+2732π2,g3=1π4−448+409602−925833/2+141292−409623π+2562+35132π2,g4=1π52336−1955842+2150553−958393−2176023π−339223+8373π2,g5=1π6−5376+3993602−4315683+63248−158722π+21762+17283π2,g6=1π74608−2949122+3149283−45840−122882π−15362+12963π2.Proof.
Letfscx=sinx/x−2+cosx/3,elx=fscx−g2x2−g3x3−g4x4−g5x5−g6x6,andeux=fscx−f1x−f2x2−f3x3−f4x4−f5x5−f6x6; then,(61)el7x=eu7x=sinc7x−13sinx=7x6−210x4+2520x2−5040sinx+−x7+42x5−840x3+5040xcosxx8−13sinx.
We lethx=7x6−210x4+2520x2−5040sinx+−x7+42x5−840x3+5040xcosx; then, we have h′x=−5x7−x8cosx<0,x∈0,π/2. Then, hx is a monotone decreasing function in 0,π/2, and we have hx≤h0=0; then, sinc7x≤0 for x∈0,π/2. Therefore, we have el7x=eu7x=fsc7x=sinc7x−1/3sinx<0.
By the definition ofelx and eux, we have(62)el0=elπ4=elπ3=elπ2=el′0=el′π4=el′π3=0,eu0=euπ4=euπ3=euπ2=eu′π4=eu′π3=eu′π2=0.
By Theorem1, there exist θj,j=1,2, such that(63)elx=fsc7θ17!x2x−π42x−π32x−π2≥0,eux=fsc7θ27!xx−π42x−π32x−π22≤0,which mean the conclusions are valid.


## 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. Theorems 2 and 3 present the novel polynomial bounds of sincx and cosx functions using the interpolation and approximation method. Theorem 4 gives the new refinements and improvements of the above five inequalities based on the results of sincx and cosx. 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 (a function). MaxErrorlow denotes the maximum error between a function and its lower bound. Similarly, MaxErrorlow 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.Table 1
Maximum errors betweensinx/x3−cosx and its bounds (Mitrinović’s inequality).
MethodErrorMaxErrorlowMaxErroruppMortici [42] (inequality (24))3.4808×10−21.4744×10−1Malesˇevic´ et al. [43] (inequality (30), n = 2)5.6770×10−45.2314×10−3The results of this paper (inequality (48))3.2409×10−83.1707×10−8Table 2
Maximum errors betweensinx/x−2+cosx/3 and its bounds (Cusa’s inequality).
MethodErrorMaxErrorlowMaxErroruppBhayo and sa´ndor [33] (inequality (6))1.1614×10−29.1985×10−3Mortici [42] (inequality (25))3.7644×10−31.9749×10−4Malesˇevic´ et al. [43] (inequality (31), n = 2)5.9603×10−61.1839×10−7The results of this paper (inequality (49))1.0803×10−81.0701×10−8The results of this paper (inequality (56))5.8244×10−73.2174×10−6Table 3
Maximum errors between2sinx/x+tanx/x and its bounds (Huygens’ inequality).
MethodErrorMaxErrorlowMaxErroruppHuygens [3] (inequality (2))7.9812×102—Chen and Cheung [37] (inequality (11))6.9173×1011.0539×10−1Chen and Cheung [37] (inequality (12))1.5550×1022.0761×10−1Chen and Paris [39] (inequality (19))6.1807×1008.3747×10−3Mortici [42] (inequality (26))2.9643×10−33.4633×102Malesˇevic´ et al. [43] (inequality (32), n = 2)1.3520×1004.4188×100The results of this paper (inequality (50))3.8724×10−77.1581×10−4Table 4
Maximum errors betweensinx/x2+tanx/x and its bounds (Wilker’s inequality).
MethodErrorMaxErrorlowMaxErroruppWilker and Porblem [4] (inequality (3))7.9826×102—Sumner et al. [36] (inequality (10))2.6424×10−26.5686×101Chen and Cheung [37] (inequality (13))1.8874×1022.4457×10−1Chen and Cheung [37] (inequality (14))1.4653×1021.9695×10−1Nenezic et al. [38] (inequality (17))3.5720×1013.9383×10−3Nenezic et al. [38] (inequality (18))5.7271×10−33.0139×10−3Chen and Paris [39] (inequality (20))2.3923×1003.9383×10−3Mortici [42] (inequality (27))8.7474×1025.5813×102Malesˇevic´ et al. [43] (inequality (33), n = 2)3.5293×1002.8103×101The results of this paper (inequality (51))3.8636×10−77.1581×10−4Table 5
Maximum errors betweenx/sinx2+x/tanx and its bounds (Wu and Srivastava’s inequality).
MethodErrorMaxErrorlowMaxErroruppWu and Srivastava [5] (inequality (4))4.6615×10−1—Chen and Cheung [37] (inequality (15))—2.1552×102Chen and Cheung [37] (inequality (16))—1.0111×102Chen and Paris [39] (inequality (21))1.0560×10−16.4478×101Wang [40] (inequality (22))2.9416×10−19.4934×10−2Mortici [42] (inequality (28))1.9612×10−1—Malesˇevic´ et al. [43] (inequality (34), m = 3)6.9338×10−29.6995×10−3The results of this paper (inequality (52))4.8271×10−83.2457×10−8Table 6
Maximum errors between3x/sinx+x/tanx and its bounds (Neuman and Sa´ndor’s inequality).
MethodErrorMaxErrorlowMaxErroruppNeuman and Sa´ndor [6] (inequality (5))7.1080×10−1—Jiang et al. [41] (inequality (23))3.2381×10−17.4286×10−2Mortici [42] (inequality (29))3.1910×10−2—Malesˇevic´ et al. [43] (inequality (35), m = 3)63.1910×10−24.1471×10−3The results of this paper (inequality (53))3.2982×10−83.2411×10−8For Mitrinović’s inequality, many researchers focused on the equivalent form of this inequality, that is, the inequality sinx/x3−cosx>0. Table 1 gives the comparison of the maximum errors between sinx/x3−cosx>0 and its bounds for different methods. In inequality (40), the maximum error between the function sinx/x3−cosx>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 sinx/x−cosx1/3 directly. Inequality (44) gives the new improvement of Mitrinović’s inequality. The maximum error between the function sinx/x−cosx1/3 and the lower bound is only 1.0834×10−8, and the maximum error of the upper bound is 6.8129×10−8.Tables1‒6 show that the results of this paper are far superior to the previous conclusions. Among the previous conclusions, only the conclusions of Malesˇevic´ 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 sinx/x−2+cosx/3 and its lower bound is 1.0803×10−8, and the maximum error between sinx/x−2+cosx/3 and its upper bound is 1.0701×10−8. The 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 Malesˇevic´ et al. [43]. Although the maximum errors of Malesˇevic´ 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. The bounds are all polynomial functions, and there are no other functions. In previous conclusions, many bounds contain not only polynomial functions but also trigonometric functions, for example,sinx,cosx,andtanx. In general, the polynomial function as the upper and lower bounds of the trigonometric function is more effective than the trigonometric function as the upper and lower bounds of the trigonometric function.At the end of this paper, we try to obtain the bounds ofsinx/x−2+cosx/3 directly using the interpolation and approximation theorem. Theorem 5 gives the new upper and lower bounds of the function sinx/x−2+cosx/3. The last row of Table 2 gives the maximum errors of inequality (56). The maximum error between the function sinx/x−2+cosx/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.



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*Source: 5595650-2021-04-15.xml*