An alternate proof of Hall's theorem on a conformal mapping inequality (Q1914927)

In this paper the authors give an alternate but simpler proof of the following theorem due to \textit{R. R. Hall} [Bull. Lond. Math. Soc. 12, 119-126 (1980; Zbl 0442.30007)]. If \(f\) is analytic for \(|z |< 1\) with \(f(0) = 0\) and is starlike of order \(1/2\). Then \[ \int^r_0 \bigl |f' (\rho e^{i \theta}) \bigr |d \rho < {\pi \over 2} \bigl |f(re^{i \theta} \bigr | \] for every \(r < 1\) and real \(\theta\).