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}?

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