 Linear coefficient bound for normalized univalent functions

For $f(z)=z+{\sum }_{n\ge 2}{a}_n{z}^n$ in the normalized univalent function class, the integral mean bound for an odd square-root transform gives $(2\pi {)}^{-1}\int |f(r{e}^{i\theta })|d\theta \le r/(1-r)$. The Cauchy integral formula for Taylor coefficients therefore yields $|a_n|\le r^{1-n}/(1-r)$. Choosing $r=(n-1)/n$ and applying the logarithm tangent bound gives the displayed bound for every $n\ge 2$.