Prove that: $\sum_{i=1}^n a_i^2\geq \sum_{i=1}^n a_ib_i$

Question

Suppose that $a_1,a_2,...,a_n$ are positive real numbers.Also assume that $b_1,b_2,...b_n$ are an arbitrary permutation of $a_i$'s. Prove that: $$\sum_{i=1}^n a_i^2\geq \sum_{i=1}^n a_ib_i.$$

We know that $\sum\limits_{i=1}^n a_i=\sum\limits_{i=1}^n b_i$ and $\sum\limits_{i=1}^n a_i^2=\sum\limits_{i=1}^nb_i^2$ . I think the rearrangement inequality can't be used to prove it.Maybe some other classic inequality should be used...

Answer 1

We know that, $\sum_{i=1}^{n}a^2_i=\sum_{i=1}^{n}b^2_i$, then,

$$\sum_{i=1}^{n}(a_i-b_i)^2=\sum_{i=1}^{n}a^2_i+\sum_{i=1}^{n}b^2_i-2\sum_{i=1}^{n}a_ib_i=2\sum_{i=1}^{n}a^2_i-2\sum_{i=1}^{n}a_ib_i\geq 0$$

Answer 2

Note that $$ \sum_{i=1}^na_i^2=\sum_{i=1}^nb_i^2$$ since the $b_i$ are a permutation of the $a_i$. Therefore it follows from the Cauchy-Schwarz inequality that $$ \sum_{i}a_ib_i\leq \Big(\sum_ia_i^2\Big)^{\frac{1}{2}}\Big(\sum_ib_i^2\Big)^{\frac{1}{2}}=\sum_ia_i^2$$

Answer 3

Your inequality it's just the Rearrangement (Chebyshov's) inequality.

Let $a_1\geq a_2\geq...\geq a_n$ and $b_1\geq b_2\geq...\geq b_n$.

Thus, for all $\sigma\in S_n$ we have $$\sum_{i=1}^na_ib_i\geq\sum_{i=1}^na_ib_{\sigma(i)}.$$

Now, take $a_i=b_i$.