Let $a=(a_1,\dots, a_n) \in \mathbb C^n$ and $b=(b_1,\dots, b_n) \in \mathbb C^n$. Suppose that $\left | \sum_{j=1}^n a_n \right | \neq 0$
I'm interested in upper bounds of the form $$ \left | \sum_{j=1}^n a_nb_n \right | \leq \left | \sum_{j=1}^n a_n \right | f(b) $$ for some function $f$. Can we obtain such bounds for a function $f$ that is independent (or just mildley dependent) of $a$?