If $\sum_{i=1}^n x_i \ge a$, then what can we know about $\sum_{i=1}^n \frac{1}{x_i}$?

Question

Suppose that $$\sum_{i=1}^n x_i \ge a$$ where $a>0$ and $x_i\in (0, b]$ for all $i$. Are there any bounding inequalities we can determine for $$\sum_{i=1}^n \frac{1}{x_i}?$$ I understand that $\sum_{i=1}^n \frac{1}{x_i} \ge \frac{n}{b}$, but I'm hoping to have some restriction that utilizes $a$. I have found this question that starts with the knowledge that $\sum_{i=1}^n x_i = a$. Following the details of the explanation in that question, I eventually conclude that $$\sum_{i=1}^n \frac{1}{x_i} \ge \frac{n^2}{\sum_{i=1}^n x_i},$$ but I'm not sure that this allows me to utilize the initial inequality $\sum_{i=1}^n x_i \ge a$. Am I missing something?