不等式问题三

数学星空

已知$x_k(1<=k<=n)$为正实数，若$x_1^2+({x_1+x_2}/2)^2+({x_1+x_2+x_3}/3)^2+...+({x_1+x_2+...+x_n}/n)^2<=alpha_n*(x_1^2+x_2^2+...+x_n^2)$求$alpha_n$ 的最小值？

数学星空

下面是有关$alpha_n=4$的经典证明：

数学星空

我们现在的问题是求更精确的解：
$alpha_n (2<=n<=10)$ ?

数学星空

我们容易算得：
$n=2$时，$alpha_2$ 是方程 $4*k^2-6*k+1=0$ 的最大实根  $alpha_2=3/4+(1/4)*sqrt(5)=1.3090169943749474241$
$n=3$时，$alpha_3$  是方程 $36*k^3-66*k^2+19*k-1=0$ 的最大实根 $alpha_3=1.49209402121373$
$n=4$时，$alpha_4$ 是方程 $576*k^4-1200*k^3+460*k^2-44*k+1=$0 的最大实根 $alpha_4=1.61871672888526$
$n=5$时，$alpha_5$ 是方程  $14400*k^5-32880*k^4+15196*k^3-2008*k^2+85*k-1=0$的最大实根，$alpha_5=1.7139329647204847012$

mathe

对于给定的n，由于是二次型，相当于求对应对称矩阵的最大特征值，所以还是比较好计算的，是一个n次方程的最大实根

wayne

数学星空 发表于 2013-10-19 22:01
我们容易算得：
时， 是方程  的最大实根  
时，  是方程  的最大实根

根据mathe的提示，算得：

1. {1,-1+x}
   
2. {2,1-6 x+4 x^2}
   
3. {3,-1+19 x-66 x^2+36 x^3}
   
4. {4,1-44 x+460 x^2-1200 x^3+576 x^4}
   
5. {5,-1+85 x-2008 x^2+15196 x^3-32880 x^4+14400 x^5}
   
6. {6,1-146 x+6568 x^2-110184 x^3+672336 x^4-1270080 x^5+518400 x^6}
   
7. {7,-1+231 x-17682 x^2+558304 x^3-7435608 x^4+38724624 x^5-65862720 x^6+25401600 x^7}
   
8. {8,1-344 x+41384 x^2-2205824 x^3+54754192 x^4-614396544 x^5+2827466496 x^6-4418426880 x^7+1625702400 x^8}
   
9. {9,-1+489 x-87168 x^2+7260328 x^3-302173200 x^4+6266941200 x^5-61458715008 x^6+255785008896 x^7-372523898880 x^8+131681894400 x^9}
   
10. {10,1-670 x+169116 x^2-20780752 x^3+1344772144 x^4-46487879136 x^5+836532818496 x^6-7348756700160 x^7+28118602828800 x^8-38569208832000 x^9+13168189440000 x^10}
    
11. {11,-1+891 x-307186 x^2+53265388 x^3-5065638336 x^4+271079242960 x^5-8088622107552 x^6+129553097587776 x^7-1037606683484160 x^8+3694930345036800 x^9-4811724352512000 x^10+1593350922240000 x^11}
    
12. {12,1-1156 x+528660 x^2-124860208 x^3+16704938800 x^4-1309222411968 x^5+60235812531648 x^6-1592408917658880 x^7+23121500548644864 x^8-171067554621296640 x^9+572283801770803200 x^10-712008517828608000 x^11+229442532802560000 x^12}
    
13. {13,-1+1469 x-869752 x^2+271855012 x^3-49416375008 x^4+5433357170800 x^5-364981350942336 x^6+14825119058046144 x^7-353819823753676032 x^8+4721684194767015936 x^9-32600616209509109760 x^10+103218762837406924800 x^11-123312192439468032000 x^12+38775788043632640000 x^13}
    
14. {14,1-1834 x+1377376 x^2-556297976 x^3+133544396128 x^4-19904201648032 x^5+1871046961217536 x^6-110650610843780736 x^7+4044593238224120064 x^8-88385128767603495936 x^9+1095642170129623965696 x^10-7116583251769378652160 x^11+21453162965717085388800 x^12-24712050750746591232000 x^13+7600054456551997440000 x^14}
    
15. {15,-1+2255 x-2111074 x^2+1079740168 x^3-334333451192 x^4+65682810276928 x^5-8352336392731680 x^6+689633532442910592 x^7-36607376825655030144 x^8+1221637108326057374976 x^9-24713439031969477309440 x^10+286993716850171384012800 x^11-1765149439656123537408000 x^12+5091241666155815116800000 x^13-5674212235766262988800000 x^14+1710012252724199424000000 x^15}
    
16. {16,1-2736 x+3145104 x^2-2002320512 x^3+783958812000 x^4-198334615265280 x^5+33185083081954048 x^6-3699457128915228672 x^7+273599353519057264896 x^8-13228098187499727114240 x^9+407563355451706996432896 x^10-7699660747367562988093440 x^11+84342242381776345984204800 x^12-493851095019926442344448000 x^13+1368495078517285886361600000 x^14-1479958528399750515916800000 x^15+437763136697395052544000000 x^16}
    
17. {17,-1+3281 x-4570688 x^2+3568618512 x^3-1736872145376 x^4+554830316155232 x^5-119366571344212736 x^6+17480738754271431424 x^7-1742424770943793510656 x^8+117143922673207347925248 x^9-5217775819992930218176512 x^10+149760479937291379917594624 x^11-2661037409297946509543669760 x^12+27651952965595203181073203200 x^13-154836508187073484816515072000 x^14+413650162487133059088384000000 x^15-435149988031383614993203200000 x^16+126513546505547170185216000000 x^17}
    
18. {18,1-3894 x+6498420 x^2-6141936800 x^3+3661753062048 x^4-1452297129787968 x^5+394000531210317952 x^6-74087341586702387712 x^7+9686466303224296930560 x^8-876267948397623956043264 x^9+54175708813403852674974720 x^10-2243433068874792895351357440 x^11+60424112600175542349957267456 x^12-1015784519584350185202790563840 x^13+10061135250128590841867619532800 x^14-54081492374003902860561481728000 x^15+139704622139801714321365401600000 x^16-143265839959268140321131724800000 x^17+40990389067797283140009984000000 x^18}
    
19. {19,-1+4579 x-9060834 x^2+10248928244 x^3-7389167226560 x^4+3585978680375136 x^5-1206494174782085056 x^6+285733838197058550400 x^7-47905670096685356338944 x^8+5676439334637869568311040 x^9-471506470711180594989986304 x^10+27056493179513817421908206592 x^11-1049434530299830833082339590144 x^12+26685045508807496796694105522176 x^13-426528468086947234919922879037440 x^14+4043167724195766580403540282572800 x^15-20931409614886249031707494187008000 x^16+52417592548474162439594233036800000 x^17-52497785617583947035588742348800000 x^18+14797530453474819213543604224000000 x^19}
    
20. {20,1-5340 x+12415132 x^2-16636752944 x^3+14341721027424 x^4-8408711594074496 x^5+3458220629748406144 x^6-1014611090955127500288 x^7+214002577736787067370752 x^8-32477517835294783873231872 x^9+3528926829268105736187177984 x^10-271676802087595498633773576192 x^11+14579193291638242406699104321536 x^12-532938282098146875857270280683520 x^13+12859317322121975442615356129280000 x^14-196253277561610511868651982159872000 x^15+1786604840795856529615771604090880000 x^16-8933266141804347050877764449075200000 x^17+21735839653125322351492139581440000000 x^18-21295064856103075198506369024000000000 x^19+5919012181389927685417441689600000000 x^20}

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Mathematica代码如下：

1. Table[{n,
   
2.   MinimalPolynomial[
   
3.    Max[x /.
   
4.      Solve[CharacteristicPolynomial[
   
5.         Last@Normal@
   
6.           CoefficientArrays[Sum[Mean[x /@ Range[i]]^2, {i, n}],
   
7.            x /@ Range[n], "Symmetric" -> True], x] == 0, x, Reals]],
   
8.    x]}, {n, 20}]

复制代码

wayne

数学星空 发表于 2013-10-19 21:00
我们现在的问题是求更精确的解：
?

数值结果就是， ：
{1,1.0000000000000000000}
{2,1.3090169943749474241}
{3,1.4920940212137277641}
{4,1.6187167288852634485}
{5,1.7139329647204847012}
{6,1.7894050727866615668}
{7,1.8514365577052729541}
{8,1.9037902036934266933}
{9,1.9488780952193906851}
{10,1.9883319111155412724}
{11,2.0233028618729224685}
{12,2.0546311393409224938}
{13,2.0829472543358262470}
{14,2.1087355204779727043}
{15,2.1323753905622218324}
{16,2.1541692657815180225}
{17,2.1743617357623476513}
{18,2.1931532170331141159}
{19,2.2107098284404712042}
{20,2.2271706772481291129}
{21,2.2426533253245527017}
{22,2.2572579517549333259}
{23,2.2710705657232905635}
{24,2.2841655167713217097}
{25,2.2966074779666640588}
{26,2.3084530286169932892}
{27,2.3197519291914983142}
{28,2.3305481571356592840}
{29,2.3408807551036892398}
{30,2.3507845306869401581}
{31,2.3602906375804299027}
{32,2.3694270613475165741}
{33,2.3782190278553193053}
{34,2.3866893486000199350}
{35,2.3948587141957536339}
{36,2.4027459450302112232}
{37,2.4103682063256928724}
{38,2.4177411934629946651}
{39,2.4248792923363557082}
{40,2.4317957186431860765}
{41,2.4385026393217601439}
{42,2.4450112787951841925}
{43,2.4513320122315501357}
{44,2.4574744476658838773}
{45,2.4634474985320004081}
{46,2.4692594479082529493}
{47,2.4749180055799048709}
{48,2.4804303588541986667}
{49,2.4858032179256121801}
{50,2.4910428564730843086}

wayne

其实 二次型对应的对称矩阵的 特征多项式 应该都是不可约的，所以6#楼的代码可以直接这样写：

1. Table[{n,Expand[n!^2CharacteristicPolynomial[Last@Normal@CoefficientArrays[Sum[Mean[x/@Range[i]]^2,{i,n}],x/@Range[n],"Symmetric"->True],x]]},{n,20}]

复制代码

数学星空

已知$x_k(1<=k<=n)$为正实数，若$x_1^2+({x_1+x_2}/2)^2+({x_1+x_2+x_3}/3)^2+...+({x_1+x_2+...+x_n}/n)^2<=beta_n*(x_1+x_2+...+x_n)^2$求$beta_n$ 的最小值？ $(n=2..20)$
Wayne可以算算看哈？

wayne

数学星空 发表于 2013-10-26 22:09
已知为正实数，若求 的最小值？
Wayne可以算算看哈？

额，被前面的误导了，跑去算各阶主子式了。

首先，各平方项系数必须为正数，所以  $\beta_n$ 最小值 不小于  $\sum_{i=1}^{n}1/i^2$

接下来，发现，取这个值的时候其他杂项系数均为正。
所以， $\beta_n$ 最小值 就是  $\sum_{i=1}^{n}1/i^2$了