Comparing hyperbolic distance with Kra's distance on the unit disk (Q436048)

Kra's distance \(d_K\) is defined on a hyperbolic Riemann surface with the aid of Teichmüller's shift mappings. The author shows that for the unit disk \(\mathbb D\), NEWLINE\[NEWLINE 2d_K<d<\frac{\pi^2}{8}e^{d_K}, NEWLINE\]NEWLINE where \(d\) is the hyperbolic metric on \(\mathbb D\) and the constants are sharp. The proof uses an explicit expression for \(d_K\) on \(\mathbb D\) in terms of complete elliptic integrals.