Geodesic
On the geometry of meromorphic functions
Sbornik. Mathematics, Tome 42 (1982) no. 2, pp. 155-196 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     title = {On~the~geometry of~meromorphic functions},
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G. A. Barsegyan. On the geometry of meromorphic functions. Sbornik. Mathematics, Tome 42 (1982) no. 2, pp. 155-196. http://geodesic.mathdoc.fr/item/SM_1982_42_2_a0/

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