An improved Schwarz lemma at the boundary (Q1738226)

Julia's classical lemma asserts that if $f$ is a holomorphic self-map of the unit disk with $f(1)=1$ (radial limit) and \[ \liminf_{z\to 1}\frac{1-|f(z)|}{1-|z|}=\beta\in\mathbb R, \] then $\beta>0$ and \[ \beta\geq\frac{|1-f(z)|^2}{1-|f(z)|^2}\;\frac{1-|z|^2}{|1-z|^2}. \] This classical result has recently been generalized or strengthened in various ways. \par The author uses another classical Schwarz-type lemma, Dieudonné's lemma, to prove the following improvement: \[ \beta\geq 2\,\frac{|1-f(z)|^2}{1-|f(z)|^2}\;\frac{1-|z|^2}{|1-z|^2}\;\frac{1-\text{Re}\left (f^*(z)\frac{1-\overline{f(z)}}{1-f(z)}\,\frac{1-z}{1-\bar{z}}\right )}{1-|f^*(z)|^2}, \] where $f^*(z)=\frac{1-|z|^2}{1-|f(z)|^2}\,f^\prime(z)$ (the hyperbolic derivative of $f$). The author also shows that the above estimate implies several of the known boundary Schwarz-type inequalities.