If $f(z)$ is an analytic function in the complex plane, $z=x+iy$, and $f(x)\neq 0$ for all $x\in \mathbb R$, does this imply that $\frac{f'(x)}{f(x)}$ is bounded on $\mathbb R$?i.e., $\big|\frac{f'(x)}{f(x)}\big|\leq C$, for some $C>0$.
Analytic in $\mathbb{C}$ implies $\left|\frac{f'(x)}{f(x)}\right|$ is bounded in $\mathbb{R}$?
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complex-analysis