A covering theorem about radial segments (Q1180315)

\textit{D. K. Blevins} [Duke Math. J. 40, 877-883 (1973; Zbl 0275.30015)] defined a \(k\)-circle to be a Jordan curve such that for every quadruple of points \(z_ 1\), \(z_ 2\), \(z_ 3\), \(z_ 4\) on it \[ X(z_ 1,z_ 2,z_ 3,z_ 4)+X(z_ 2,z_ 3,z_ 4,z_ 1)\leq k^{-1} \] with \(0<k\leq 1\), \[ X(z_ 1,z_ 2,z_ 3,z_ 4)=q(z_ 1,z_ 2)q(z_ 3,z_ 4)q(z_ 1,z_ 3)^{-1}q(z_ 2,z_ 4)^{-1} \] with \(q\) chordal distance. He gave some covering theorems [Can. J. Math. 28, 627- 631 (1976; Zbl 0362.30013)] for functions \(f\) in the family \(S\) which map the unit disc \(U\) onto a domain bounded by a \(k\)-circle. The family of these is denoted by \({\mathfrak S}_ k\). In particular if \(\ell(\varphi)\) denotes the linear measure of the intersection of \(f(U)\) with the ray of argument \(\varphi\) and \(d=\pi(4(\pi-\sin^{-1}k))^{-1}\) he proved for \(f\in{\mathfrak S}_ k\), \[ \ell(\varphi)\ell(-\varphi)\geq 4d^ 2\sin^ 2\varphi,\quad \pi/6\leq\varphi\leq\pi/2, \qquad \ell(\varphi)\ell(- \varphi)\geq d^ 2,\quad 0\leq\varphi<\pi/6, \] with a corresponding equality statement. In the same context the author obtains the following result: \[ \ell(\theta)^{-1}+\ell(\pi-\theta)^{-1}\leq(d \cos \theta)^{-1},\;0\leq\theta\leq\pi/3, \quad \ell(\theta)^{- 1}+\ell(\pi-\theta)^{-1}\leq 2d^{-1},\;\pi/3<\theta\leq\pi/2, \] with a corresponding equality statement.