Say $(a_1, a_2, …, a_n)$ and $(b_1, b_2, …, b_n)$ are two sequences of increasing or decreasing positive real numbers.
Now, suppose we replace $(b_1, b_2, …, b_n)$ by $(a_1^{-1}, a_2^{-1}, … , a_n^{-1})$.
By Cauchy-Schwarz or AM-HM,
$$ \left(\sum_{i=1}^n{a_i}\right)\left(\sum_{i=1}^n{a_i^{-1}}\right) \geq n^2 $$
In contrast, Chebyshev Inequality affirms that$-$
$$ \left(\sum_{i=1}^n{a_i}\right)\left(\sum_{i=1}^n{a_i^{-1}}\right) \leq n^2 $$
I found this result while playing around with Inequalities, and now I'm stuck.
How can this happen that one standard result states some other false?
I know that the first Inequality is definitely true, but what happens with the second one?
Answers are appreciated. Thanks!
Best Answer
Chebyshev's sum inequality states that $$ {1 \over n}\sum _{{k=1}}^{n}a_{k}\cdot b_{k}\geq \left({1 \over n}\sum _{{k=1}}^{n}a_{k}\right)\left({1 \over n}\sum _{{k=1}}^{n}b_{k}\right) $$ if $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ are both increasing (or both decreasing), and the reverse inequality $$ {1 \over n}\sum _{{k=1}}^{n}a_{k}\cdot b_{k}\leq \left({1 \over n}\sum _{{k=1}}^{n}a_{k}\right)\left({1 \over n}\sum _{{k=1}}^{n}b_{k}\right) $$ if the sequences are in opposite order (one increasing and the other decreasing).
With $(b_1, b_2, ..., b_n) =(a_1^{-1}, a_2^{-1}, ... , a_n^{-1})$ and increasing (or decreasing) $a_j$ we have the second case, which gives $$ 1 \le \left({1 \over n}\sum _{{k=1}}^{n}a_{k}\right)\left({1 \over n}\sum _{{k=1}}^{n}\frac{1}{a_{k}}\right) $$ i.e. the same result as Cauchy-Schwarz or the AM-HM inequality.