Past exam of the complex analysis: conformal maps and geometry course of the University of Oxford / 2023 / 1 / b / v / Solution

Use $h(\sqrt{r}{e}^{i\theta }{)}^2=\varphi (r{e}^{2i\theta })$. As $\theta$ runs around the circle, the doubled angle goes around it twice, so iv givesThe Cauchy integral formula for the Taylor coefficients now impliesFor $n\ge 2$, choosing $r=(n-1)/n$ gives the linear coefficient bound for normalized univalent functions,The last inequality follows from $\log (1+x)\le x$ for $x>-1$.