Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 13F / d / Solution

No. The nonvanishing holomorphic functions $f_n(z)=e^{nz}$ satisfy $f_n(0)=1$, butAvoiding 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.