Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 2 / 12 / 12.2A / b / ii

If $h^{\prime }(0)=0$, let $a_k{z}^k$ be the first nonzero Taylor term. Then $k\ge 2$ and $a_k$ is real. But $\mathrm{Im}(a_k{r}^k{e}^{ik\theta })$ changes sign as $0<\theta <\pi$, and for small $r$ the leading term controls the sign. This contradicts positivity in the upper half-plane. Hence $h^{\prime }(0)\ne 0$.