Bestness of the parabola theorem for continued fractions (Q1196870)

The author proves the following theorem: Let \(b\) be a complex number such that \[ b\notin P_ \alpha=\{z\in\mathbb{C};| z|-\text{Re}(ze^{- i2\alpha})\leq 1/2\cos^ 2\alpha\}, \] where \(\alpha\) is a fixed real number in \(\langle -\pi/2,\;\pi/2\rangle\). Then there exists a point \(a\) on the boundary \(\partial P_ \alpha\) of \(P_ \alpha\) such that the linear fractional transformation \[ S(w)={\hbox to 1.5em{\hfil\(a\)\hfil}|\over|\hbox to 1.5em{\hfil1\hfil}}+{\hbox to 2.25em{\hfil\(b\)\hfil}|\over|\hbox to 2.25em{\hfil\(1+w\)\hfil}}={a(1+w)\over 1+b+w} \] is elliptic.