Is there Inequality relating the arithmetic mean (AM) and the dot product

Question

Is there an inequality of the kind:

$$ \sum_{i=1}^n a_{i}b_{i} \geq C_{1}\left(\frac{\sum_{i=1}^{n} a_{i}}{n}\right)C_{2}\left(\frac{\sum_{i=1}^{n} b_{i}}{n}\right), $$

i.e., one relating the dot (inner) product $\sum\limits_{i} a_{i}b_{i}$ and the arithmetic means for $A$ and $B$, where the dot product is the greater quantity?

Here, $C_1$ and $C_2$ might not necessarily be constants, but possibly involving the maximum or minimum of $A$ and/or $B$ or something else.

Well-known inequalities like the Rearrangement inequality and the Holder's inequality do not give what I want (neither the Holder’s reverse inequality).

Edit after the comment of @Jack D'Aurizio: the dot product is not $0$.

Answer 1

If $0 < m_1 \le a_i \le M_1$ and $0 < m_2 \le b_i \le M_2$ for all $i$, then $$\sum_{i=1}^n a_ib_i \ge n\left(1 - \frac{(M_1 - m_1)(M_2 - m_2)}{4\sqrt{M_1m_1M_2m_2}}\right)\left(\frac{1}{n}\sum_{i=1}^n a_i\right)\left(\frac{1}{n}\sum_{i=1}^n b_i\right).$$

[1] Dragomir and Khan, Two Discrete Inequalities of Grüss Type Via Pólya-Szegö and Shisha Results for Real Numbers.