A~bound of the exterior arcs for a~univalent mapping

Yu. A. Litvinchuk ; I. M. Milin

Matematičeskie zametki, Tome 18 (1975) no. 3, pp. 367-378

Résumé

In this paper we consider the intersection of the circle $|w|=x$ with the image of the disc $|z|\le r$, $0$, under the mapping of a function of the form $f(z)=z+c_2z^2+\dots$ which is univalent analytic in $|z|1$. Earlier I. E. Bazilevich proved that for $x\ge e^{\pi/e}r$ the measure of the above intersection does not exceed the measure of the intersection produced by the function $f^*(z)=\frac z{(1-\eta z)^2}$, $|\eta|=1$. In this paper I. E. Bazilevich's ideas are used to strengthen some of his results.