A bound of the exterior arcs for a univalent mapping

In this paper we consider the intersection of the circle $|w|=x$ with the image of the disc $|z|\le r$, $0<r<1$, 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.