How find this inequality minimum $\sum_{i=1}^{n}a^2_{i}-2\sum_{i=1}^{n-1}a_{i}a_{i+1}$

Question

Let $n$ be a given positive integer, and let $a_{1},a_{2},\cdots,a_{n}\ge 0$ such that $a_{1}+a_{2}+\cdots+a_{n}=1$. Find this minimum value $$a^2_{1}+a^2_{2}+\cdots+a^2_{n}-2a_{1}a_{2}-2a_{2}a_{3}-2a_{3}a_{4}-\cdots-2a_{n-2}a_{n-1}-2a_{n-1}a_{n}.$$

I think maybe we can use this well known $$a^2_{1}+a^2_{2}+\cdots+a^2_{n}\ge a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n-1}a_{n}+a_{n}a_{1}?$$

But this problem is only $$-2a_{1}a_{2}-2a_{2}a_{3}-2a_{3}a_{4}-\cdots-2a_{n-2}a_{n-1}-2a_{n-1}a_{n}$$ so I can't it.Thank you for help.

By the way: I don't know this problem have Someone research?if No,I think this is interesting problem.

Answer 1

I show below that the minimum is $-\frac{1}{6}$ for $n \geq 4$, and $-\frac{1}{7}$ when $n = 3$.

If we put

$$ Q_n(a_1,a_2,\ldots ,a_n)= a^2_{1}+a^2_{2}+\cdots+a^2_{n}-2a_{1}a_{2}-2a_{2}a_{3}-2a_{3}a_{4}-\cdots-2a_{n-2}a_{n-1}-2a_{n-1}a_{n} $$

and

$$ R_n(a_1,a_2,\ldots ,a_n)=Q_n(a_1,a_2,\ldots,a_n)+\frac{(a_1+a_2+a_3+\ldots +a_n)^2}{6} $$

then for $n\geq 4$, we have the identity

$$ R_n(a_1,a_2,a_3,\ldots,a_n)= \frac{1}{42}\bigg(\sum_{k=1}^{n-2}a_k-5a_{n-1}+7a_n\bigg)^2 +\frac{1}{28}\bigg(\sum_{k=1}^{n-3}2a_k-5a_{n-2}+4a_{n-1}\bigg)^2 +\frac{1}{4}\bigg(\sum_{k=1}^{n-4}2a_k-2a_{n-3}+a_{n-2}\bigg)^2 $$

Since $Q_n(0,\ldots, 0,\frac{1}{6},\frac{1}{3},\frac{1}{3},\frac{1}{6})=-\frac{1}{6}$, we see that the minimum is $-\frac{1}{6}$ for $n\geq 4$.

For $n=3$, the minimum is $-\frac{1}{7}$ attained at $(\frac{2}{7},\frac{3}{7},\frac{2}{7})$ (thanks Macavity) because of

$$ Q_3(a_1,a_2,a_3)+\frac{(a_1+a_2+a_3)^2}{7}= \frac{1}{56}\bigg(a_1-6a_2+8a_3\bigg)^2 +\frac{1}{8}\bigg(-3a_1+2a_2\bigg)^2 $$

Answer 2

You almost did the problem yourself, just left out a few bits:

Assume, without loss of generality,

$$a_1 \geq a_2 \geq \dots \geq a_n$$

By the rearrangement inequality,

$$\implies a_1^2 + a_2^2 \dots a_n^2 \geq a_1a_2 + a_2a_3 \dots a_{n-1}a_n + a_na_1$$

Clearly, the required will be minimum if equality holds true above.

This will happen when

$$a_1 = a_2 = a_3 = \dots = a_n = a$$

This means, the maximum value of the inequality is:

$$\sum_{i = 1}^n a^2 - 2a^2 = -na^2$$

Since, from the first equation,

$$na = 1$$

$$a = \frac{1}{n}$$

Therefore, the minimum is,

$$-n\cdot\left(\frac{1}{n^2}\right) = -\frac{1}{n}$$