Über Längen- und Flächenverzerrungen für die Carathéodorysche Klasse. (On length and areal distortions for the Carathéodory class) (Q1078715)

Let P(\(\alpha)\) denote the Carathéodory class of order \(\alpha <1\) consisting of analytic functions p holomorphic in the unit disk which are normalized by \(p(0)=1\) and satisfy Re p\(>\alpha\). Let F(\(\alpha)\) be the class consisting of \(f=zp\) with \(p\in P(\alpha)\). An additive family of operators \(\{L^{\lambda}\}\) depending on a continuous parameter \(\lambda\geq 0\) is generated by a probability measure \(\sigma\) supported on the unit interval [cf. \textit{Y. Komatu}, Bull. Fac. Sci. Eng., Chuo Univ., Ser. I 22, 1-22 (1979; Zbl 0462.30023)]. In the present paper distortion inequalities concerning the length and area of the concentric circumference and disk, respectively, by the mapping \[ w=f_{\lambda}(z)/z\quad with\quad f_{\lambda}=L^{\lambda}f,\quad f\in F(\alpha). \] Special reference is made to a particular case of the family \(\{L^{\lambda}\}\) generated by \(\sigma (t)=t\).