Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 3 / 3E (Further Analysis)

Fix any $z_0\in \mathbb{C}$. The Cauchy estimate on the circle $|z-z_0|=R$ applies for every $R>0$ because $f$ is an entire function. It givesTherefore $f^{\prime }(z_0)=0$. Since $z_0$ was arbitrary and $\mathbb{C}$ is connected, $f$ is constant. This is the sublinear case of the polynomial growth theorem for entire functions.

The composition $g(z)=e^{-f(z)}$ is an entire function, andBy Liouville's theorem, $g$ is constant. Since the exponential function never vanishes,Thus $f$ is constant. Notice that constancy of an exponential alone need not be inverted by choosing a global logarithm; differentiating supplies the conclusion directly.