Abstract:
This paper establishes various propositions characterizing the geometric behavior of meromorphic functions $w(z)$ in $|z|<\infty$. “Distortion” theorems for these functions form a basis for the arguments. Namely, a finite number of nice curves $\Gamma_\nu$, $\nu=1,2,\dots,q$, in the $w$-plane are considered (in particular, $\Gamma_1$ may be a straight line) and information is obtained about the lengths $L(r, \Gamma_\nu)$ of the sets $w^{-1}(\Gamma_\nu)\cap\{z:|z|\le r\}$, $\nu=1,2,\dots,q$. Qualitatively, the main result is as follows: on some sequence $r_n\to\infty$ \begin{equation}
\sum^q_{\nu=1}L(r, \Gamma_\nu)\le KrA(r),
\tag{1}
\end{equation}
where $K$ is an absolute constant, and $A(r)$ is the Ahlfors characteristic.
Bibliography: 29 titles.
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