On a strengthened Schwarz-Pick inequality (Q1301888)

Let \(\Delta=\{z\mid |z|<1\}\) be the unit disk in \({\mathbb C}\), and let \(\varphi_w(z)=\frac{w-z}{1-\overline w z}\) be a Möbius transformation. The hyperbolic distance \(\rho\) is defined by \(\rho(z,w)= \frac 12\log{(1+|\varphi_w(z)|)/(1-|\varphi_w(z)|)}\), \(w,z\in\Delta\). As is well known the Schwarz-Pick lemma says for an analytic function \(f:\Delta\to\Delta\) that \(\rho(f(z),f(w))\leq \rho(z,w)\) for all \(z,w\in\Delta\). The author sharpens this inequality by showing \[ \rho(f(z),f(w))\leq \rho(z,w)+\tfrac 12\log \Biggl[1-(1-A)\frac{2|\varphi_z(w)|}{(1+|\varphi_z(w)|)^2} \Biggr] \quad\text{for all } z,w\in\Delta, \] where \(A\) is between \(0\) and \(1\). Further, using this result Julia's lemma is strengthened.