不等式难题

数学星空 · 发表于 2015-6-21 08:08:50

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1.若$a,b,c \gt 0 $ 且$a+b+c=1$,求\[\frac{1}{a+b^2}+\frac{1}{b+c^2}+\frac{1}{c+a^2}\]的最小值？

2.若$a,b,c,k \gt 0 $ 且$a+b+c=1$,求\[\frac{1}{a+b^k}+\frac{1}{b+c^k}+\frac{1}{c+a^k}\]的最小值？

3.若$a_i (i=1..n),k \gt 0 $ 且$a_1+a_2+...+a_n=1$,求\[\frac{1}{a_1+a_2^k}+\frac{1}{a_2+a_3^k}+...+\frac{1}{a_n+a_1^k}\]的最小值？

4.若$a_i (i=1..n),k \gt 0 $ ,$s \in R$ 且$a_1+a_2+...+a_n=1$,求\[\frac{1}{(a_1+a_2^k)^s}+\frac{1}{(a_2+a_3^k)^s}+...+\frac{1}{(a_n+a_1^k)^s}\]的最小值？

5.若$a_i (i=1..n),k \gt 0 $ ,$s,t \in R$ 且$a_1+a_2+...+a_n=1$,求\[\frac{1}{(a_1^t+a_2^k)^s}+\frac{1}{(a_2^t+a_3^k)^s}+...+\frac{1}{(a_n^t+a_1^k)^s}\]的最小值？

数学星空 · 发表于 2015-6-21 11:08:49

对于第1问：

仅当$a,b,c$取下列方程的实根：

$774144x^{27}-6967296x^{26}+32901120x^{25}-106334208x^{24}+270355968x^{23}-583690752x^{22}+1147745664x^{21}-2127342528x^{20}+3708075024x^{19}-5927352480x^{18}+8496262296x^{17}-10770428544x^{16}+11979589941x^{15}-11621888361x^{14}+9773328369x^{13}-7065521081x^{12}+4359747743x^{11}-2288408071x^{10}+1044657625x^9-448146989x^8+208533820x^7-109307100x^6+55399246x^5-23049242x^4+6935253x^3-1346057x^2+146935x-6767=0$

${a = 0.1814495608, c =0 .3529465026, b =0 .4656039366}$

取最小值$s=6.797025704$,满足下列方程

$4227136s^9-51889328s^8+194004672s^7-239831432s^6-119857988s^5+400521285s^4+44318448s^3-279928224s^2-117441792s-13886208=0$

发疯的_小男孩 · 发表于 2015-6-22 10:51:40

对于第一个，最小值是下列方程的唯一实根
${k}^{7}-{\frac {34\,{k}^{6}}{3}}+{\frac {109\,{k}^{5}}{3}}-{\frac {109
\,{k}^{4}}{3}}+51\,{k}^{3}-{\frac {440\,{k}^{2}}{3}}-75\,k-{\frac {125
}{12}}
$
在这个区间内：$[{\frac {895769052971}{137438953472}},{\frac {223942263243}{
34359738368}}]
$
约等于：6.517577662

发疯的_小男孩 · 发表于 2015-6-22 10:54:08

∑1/(a+b2)≥27/4 . (a+b+c = 1)

kastin · 发表于 2015-6-22 18:58:37

Minimize[{1/(a + b^2) + 1/(b + c^2) + 1/(c + a^2),
a + b + c == 1 && a > 0 && b > 0 && c > 0}, {a, b, c}]

复制代码

Minimize::wksol: 警告：在定义目标函数并且满足约束条件的区域内，不存在最小值；返回边界上的一个结果。

倪举鹏 · 发表于 2015-6-24 10:01:53

eq := 1/(y^2+x)+1/(y+(1-x-y)^2)+1/(x^2-x-y+1);
print(`output redirected...`); # input placeholder
1 1 1
------ + ---------------- + --------------
2 2 2
y + x y + (1 - x - y) x - x - y + 1
solve({diff(eq, x) = 0., diff(eq, y) = 0.}, {x, y});
print(`output redirected...`); # input placeholder
{x = 0.3333333333, y = 0.3333333333},
{x = 0.3529465026, y = 0.1814495608},
{x = 0.4656039366, y = 0.3529465026},
{x = 0.1814495608, y = 0.4656039366}, {
x = -1.200715075 + 1.240036780 I,
y = 0.2308887916 + 0.009963391867 I}, {
x = -0.1956168598 - 1.022562259 I,
y = 1.085991006 + 0.3422484039 I}, {
x = 0.2308887916 - 0.009963391867 I,
y = 1.969826283 + 1.250000172 I}, {
x = -0.4007763176 - 0.2732427364 I,
y = 0.8919184676 + 0.9490297519 I}, {
x = 0.8919184676 - 0.9490297519 I,
y = 0.5088578500 + 0.6757870154 I}, {
x = 0.4259628796 - 1.668925044 I,
y = 0.3031052782 + 0.7396409128 I}, {
x = 0.3031052782 + 0.7396409128 I,
y = 0.2709318421 + 0.9292841308 I}, {
x = 0.2709318421 - 0.9292841308 I,
y = 0.4259628796 + 1.668925044 I}, {
x = 1.085991006 + 0.3422484039 I,
y = 0.1096258536 + 0.6803138552 I}, {
x = 0.1096258536 - 0.6803138552 I,
y = -0.1956168598 + 1.022562259 I}, {
x = 1.969826283 - 1.250000172 I,
y = -1.200715075 + 1.240036780 I}, {
x = 0.5088578500 + 0.6757870154 I,
y = -0.4007763176 + 0.2732427364 I}, {
x = 0.5088578500 - 0.6757870154 I,
y = -0.4007763176 - 0.2732427364 I}, {
x = 1.969826283 + 1.250000172 I,
y = -1.200715075 - 1.240036780 I}, {
x = 0.1096258536 + 0.6803138552 I,
y = -0.1956168598 - 1.022562259 I}, {
x = 1.085991006 - 0.3422484039 I,
y = 0.1096258536 - 0.6803138552 I}, {
x = 0.2709318421 + 0.9292841308 I,
y = 0.4259628796 - 1.668925044 I}, {
x = 0.3031052782 - 0.7396409128 I,
y = 0.2709318421 - 0.9292841308 I}, {
x = 0.4259628796 + 1.668925044 I,
y = 0.3031052782 - 0.7396409128 I}, {
x = 0.8919184676 + 0.9490297519 I,
y = 0.5088578500 - 0.6757870154 I}, {
x = -0.4007763176 + 0.2732427364 I,
y = 0.8919184676 - 0.9490297519 I}, {
x = 0.2308887916 + 0.009963391867 I,
y = 1.969826283 - 1.250000172 I}, {
x = -0.1956168598 + 1.022562259 I,
y = 1.085991006 - 0.3422484039 I}, {
x = -1.200715075 - 1.240036780 I,
y = 0.2308887916 - 0.009963391867 I}
a := .3529465026;
print(`output redirected...`); # input placeholder
0.3529465026
b := .1814495608;
print(`output redirected...`); # input placeholder
0.1814495608
1/(b^2+a)+1/(b+(1-a-b)^2)+1/(a^2-a-b+1);
print(`output redirected...`); # input placeholder
6.797025704

复制代码

数学星空 · 发表于 2015-6-25 20:20:54

的确，如kastin 所述：

当$a=0.3549184773,b=0,c=1-a=0.6450815227$时取最小值$s=6.517577662$

并且$a$是下列方程的实根

$a^7-a^6+a^5-3a^4+9a^3-9a^2+5a-1=0$

$s$是下列方程的实根

$12s^7-136s^6+436s^5-436s^4+612s^3-1760s^2-900s-125=0$

关于第二问：$\frac{1}{a+b^k}+\frac{1}{b+c^k}+\frac{1}{c+a^k}$的最小值？

仅当$b=0,c=1-a$时取得最小值．

$sa(1-a)^ka^k-sa^2(1-a)^k+sa(1-a)^k-(1-a)^ka^k-aa^k+a^2-(1-a)^k-a＝0$

$2sa^2(1-a)^kka^k-sa^3(1-a)^kk+sa^2(1-a)^ka^k-sa(1-a)^kka^k-2sa^3(1-a)^k+sa^2(1-a)^kk-2(1-a)^kka^ka-sa(1-a)^ka^k+3sa^2(1-a)^k-a^kka^2+(1-a)^kka^k-(1-a)^kka-sa(1-a)^k-a^2a^k+a^kka+2a^3+aa^k-3a^2+a＝0$

例如：

$n=3$时

当$a = 0.2813275135, b = 0, c =1－a=0.7186724865$时取最小值$ s = 7.598274143$

并且$a$是下列方程的实根

$a^{10}-4a^9+4a^8-6a^7+5a^6-a^4-8a^3+11a^2-6a+1=0$

$s$是下列方程的实根

$621s^{10}-1458s^9+3969s^8-227178s^7+435672s^6-2620424s^5-1207505s^4-1392046s^3-44705s^2-75050s+51125=0$

$n=4$时

当$a = 0.2398564113, b = 0, c =1－a= 0.7601435887$时取最小值$ s = 8.474140875$

并且$a$是下列方程的实根

$a^{13}-5a^{12}+10a^{11}-4a^{10}-3a^9+9a^8-8a^7-4a^6+19a^5-23a^4+22a^3-16a^2+7a-1=0$

$s$是下列方程的实根

$58624s^{13}+1093120s^{12}+44226304s^{11}+78768640s^{10}-3164587776s^9-13403581056s^8-2816774656s^7-14580377408s^6-7334611392s^5-1928812096s^4-3667394112s^3+19221736s^2+376547976s-171779897=0$

$n=5$时

当$a =0.2113929122, b = 0, c =1-a=0.7886070878$时取最小值$ s = 9.276569546$

并且$a$是下列方程的实根

$a^{16}-6a^{15}+15a^{14}-20a^{13}+13a^{12}-22a^{11}+34a^{10}-30a^9+15a^8+4a^6-34a^5+50a^4-40a^3+22a^2-8a+1=0$

$s$是下列方程的实根

$8965625s^{16}-720450000s^{15}+139000006250s^{14}-3629806156250s^{13}+219225969213125s^{12}-2048711947753500s^{11}+2504098341604775s^{10}-4198399467942936s^9+229034262401362s^8-1911698414867680s^7-697316710920623s^6-2164417173747892s^5+96235446985043s^4-32652158148216s^3+87048362883455s^2-91995019228846s+27687171314981=0$

数学星空 · 发表于 2015-6-26 18:23:02

对于下式

$F(a,b,c,d) =\frac{1}{a+b^2}+\frac{1}{b+c^2}+\frac{1}{c+d^2}+\frac{1}{d+a^2}$

做代换$b=t,c=c-t$得到 ($0\leqslant t \leqslant c$)

$F(a,t,c-t,d)=\frac{1}{a+t^2}+\frac{1}{t+(c-t)^2}+\frac{1}{c-t+d^2}+\frac{1}{d+a^2}$

易证：$F(a,0,0,1-a) \leqslant F(a,0,c,d) \leqslant F(a,t,c-t,d)$

即对于第3问题及第4问题，第5问题，猜测取最小值条件为：$a_2=a_3=...=a_{n-1}=0$

也就转化为第2问题

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