Is there an inequality for the product of two sequences, where one contains the term $a_i$ and the other its reciprocal $1/a_i$? In my work I have come across the following product of summations:
$$
\left( \sum_{i=1}^n a_i b_i c_i \right) \left( \sum_{j=1}^n \frac{b_j}{a_j} \right),
\qquad\qquad (1)
$$
where $a_i, c_i \in \mathbb{R}_{> 0}, b_i \in \mathbb{R}$.
Is there an inequality I can use to get a better lower bound than
$$
\left( \sum_{i=1}^n a_i b_i c_i \right) \left( \sum_{j=1}^n \frac{b_j}{a_j} \right)
\geq
\frac{\min(a_i)}{\max(a_i)} \left( \sum_{i=1}^n b_i c_i \right) \left( \sum_{j=1}^n b_j \right)?
$$
I've been wrangling with this problem for a few days, and Google hasn't been much help. I am not a mathematician by training so this may be because I simply don't know how to phrase the right question.
Many thanks in advance!
(Something in my gut tells me this should be true:
$$
\left( \sum_{i=1}^n a_i b_i c_i \right) \left( \sum_{j=1}^n \frac{b_j}{a_j} \right)
\geq
\left( \sum_{i=1}^n b_i c_i \right) \left( \sum_{j=1}^n b_j \right);
$$
is it?)

Edit
Thanks to commenters for pointing out that both of my inequalities were false. I'll add some more details to hopefully clarify where my question comes from.
This question derives from trying to find the sign of the following expression:
$$
- \sum_{i=1}^n x_i y_i z_i^2 + \left( \sum_{i=1}^n x_i z_i \right)\left( 
\sum_{j=1}^n x_j y_j z_j \right)
\qquad\qquad (2)
$$
where $x_i, y_i \in \mathbb{R}_{\geq 0}, z_i \in \mathbb{R}$, and $\sum_{i=1}^n x_i = 1$.
I used the Cauchy-Schwartz inequality on the first term to get:
$$
\left(\sum_{j=1}^n x_j\right) \left( \sum_{i=1}^n x_i y_i z_i \right)
\geq
\left( \sum_{i=1}^n x_i \sqrt{y_i} z_i \right)^2,
\qquad\qquad (3)
$$
so if $y_i$ is the same for all $i$, the expression $(2)$ is negative semidefinite. (To get the original expression (1) I assumed that $y_i \neq 0$ and multiplied (3) through by $\tfrac{\Pi_{i=1}^{n}\sqrt{y_i}}{\Pi_{i=1}^{n}\sqrt{y_i}}$, defining $a_i = \Pi_{j \neq i}\sqrt{y_i}, b_i = y_i, c_i = x_i z_i$.)
My question is: can something be said about the sign of $(2)$ for general $y_i \in \mathbb{R}_{\geq 0}$?
Thank you!
 A: Here are my two cents, I suspect that you cannot in general do much better. Here is a cooked up toy example: Let $a_1 = 1, a_2 = 100$, $b_1=1, b_2=100$ and $c_1=100^2, c_2=1$. Then
$$ \left(\sum a_ib_ic_i\right)\left(\sum b_i/a_i \right) =(100^2 + 100^2) \cdot (1+1) = 40000, $$
and
$$ \left( \sum b_ic_i \right) \left( \sum b_i\right) = (100^2+100)\cdot(1+100) = 1020100. $$
This shows in particular that your gut feeling in this case is incorrect, as the second sum is (quite a bit) larger than the first. Moreover,
$$\frac{\min (a_i)}{\max (a_i)} \left( \sum b_ic_i \right) \left( \sum b_i\right) = \frac{1}{100}1020100 = 10201.$$
This is roughly $1/4$th of the first sum, so we conclude that
$$ \left(\sum a_ib_ic_i\right)\left(\sum b_i/a_i \right)  \geq 4 \frac{\min (a_i)}{\max (a_i)} \left( \sum b_ic_i \right) \left( \sum b_i\right)  $$
does not hold in general.
This is just an exploratoty toy example, I think it is likely that by choosing the numbers in a smarter manner we can reduce the $4$ down, but not necessarily all the way to $1$.
