Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 2 / 12B / b

Liouville theorem states that every bounded entire function is constant.

Because $\mathbb{C}$ is simply connected, the harmonic function $u$ has a global harmonic conjugate $v$, sois entire. Positivity givesLiouville's theorem makes $e^{-F}$ constant. Differentiating this nonzero constant gives $F^{\prime }=0$, so $F$, and hence $u$, is constant.

SetThen $g$ is entire and never takes a real value. The continuous function $\mathrm{Im}g$ is nowhere zero; because $\mathbb{C}$ is connected, it has one sign everywhere. Thus either $\mathrm{Im}g$ or $-\mathrm{Im}g$ is a positive harmonic function. Part (i) makes it constant. The Cauchy-Riemann equations then force the real part of $g$ to be constant as well. Hence $g$ and therefore $f$ are constant.