Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 12B / b / iii / Solution

Write $h(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$. Cauchy's coefficient formula on $|z|=R$ givesExcept at the two measure-zero points where $\cos \theta =0$, the hypothesis givesSince $|\cos \theta {|}^{-1/2}$ is integrable,for a constant $C$ independent of $R$. Letting $R\to \infty$ shows that every $a_n=0$. Hence $h\equiv 0$, in particular $h$ is constant. This proves the entire function under a horizontal inverse-square-root bound result.