求助：各 ai > 0, 则 a1/(a2+a3)+a2/(a3+a4) + ... + an/(a1+a2) ≧ n/2 ？？？

uk702

我们知道，史上著名不等式： \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}≥\frac{3}{2}\) 。


那么，一般地，若 \(a_1, a_2, ...，a_n\) > 0, 则 \(\frac{a_1}{a_2+a_3}+\frac{a_2}{a_3+a_4}+...+\frac{a_n}{a_1+a_2} ≧ \frac{n}{2} \) 可成立？

求助大家，是否有这方面的资料？

王守恩

我喜欢把答案先找出来，慢慢琢磨为什么会是这样？想不通就先摆一摆(不要有负担)。
Minimize[{a/(b + c) + b/(c + a) + c/(a + b), a > 0, b > 0, c > 0}, {a, b, c}]
{3/2, {a -> 1, b -> 1, c -> 1}}
NMinimize[{a/(b + c) + b/(c + d) + c/(d + a) + d/(a + b), a > 0,  b > 0, c > 0, d > 0},
{2., {a -> 2.30637, b -> 0.732111, c -> 2.30637, d -> 0.732111}}
NMinimize[{a/(b + c) + b/(c + d) + c/(d + e) + d/(e + a) + e/(a + b), a > 0, b > 0, c > 0,
{2.5, {a -> 0.734372, b -> 0.734372, c -> 0.734372, d -> 0.734372, e -> 0.734372}}
NMinimize[{a/(b + c) + b/(c + d) + c/(d + e) + d/(e + f) + e/(f + a) +f/(a + b), a > 0, b > 0,
{3., {a -> 1.44065, b -> 0.808194, c -> 1.44065, d -> 0.808194, e -> 1.44065, f -> 0.808194}}
NMinimize[{a/(b + c) + b/(c + d) + c/(d + e) + d/(e + f) + e/(f + g) +f/(g + a) + g/(a + b),
{3.5, {a -> 1.72387, b -> 1.72387, c -> 1.72387, d -> 1.72387, e -> 1.72387, f -> 1.72387,
NMinimize[{a/(b + c) + b/(c + d) + c/(d + e) + d/(e + f) + e/(f + g) + f/(g + h) + g/(h + a) +
{4., {a -> 0.947866, b -> 1.40559, c -> 0.947866, d -> 1.40559, e -> 0.947866, f -> 1.40559,

uk702

真没想到，这么简单的题目竟然如此之难！且结论是当 n 充分大时并不成立的。下面的内容来自 天山草@ 的帖子 /bbs/forum.php?mod=viewthread&tid=2045138&extra=page%3D1