Prove that a holomorphic function with postive real part is constant
If you can show that $e^{-f(z)}$ is constant, then: $\frac {d}{dz}e^{-f(z)}=e^{-f(z)}f'(z)=0$, so $f'(z)=0$ , and $f(z)$ is constant.
If you can show that $e^{-f(z)}$ is constant, then: $\frac {d}{dz}e^{-f(z)}=e^{-f(z)}f'(z)=0$, so $f'(z)=0$ , and $f(z)$ is constant.