## a question about compex analysis

Let $f(z)=\sum_{j \neq 0} c_j z^j$ be a meromorphic function(note, i.e.  $c_0=0$) . Ask: $\max_{|z|=r} |1+f(z)|$  $\geq 1$ for all r?

Remark, it can be shown that if r is small enough, then the statement is true, basically by argument principle implies $\{1+f(z) \| |z|=r \text{ for small r}\}$ will surrounding 1, hence it has a point with norm larger than 1. So, my question is is it true for all $r \in \mathbb{C}$?