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

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.