In the book Sadri Hassani for mathematical physics, the author derives the following result:-
I am not sure how the conclusion that the term $\frac{f'(z)}{f(z)}$ is bounded follows. For instance, I can take $f(z) = e^{z^2}$ and the term becomes $2z$, which is clearly unbounded. Is there a problem in the proof or am I understanding something incorrectly?