Solution
= Solution
\b[No.] The nonvanishing holomorphic functions $f_n(z)=e^{nz}$ satisfy $f_n(0)=1$, but
$$
|f_n(1/2)|=e^{n/2}\longrightarrow\infty.
$$
Avoiding zero alone permits an unbounded <holomorphic logarithm>'s real part; the omitted ray in part (c) was what constrained the image after taking a square root.