Applications of the principle of partitioning of meromorphic functions. I: Comparability property (Q1129746)

For \(z\in\mathbb{C}\) we denote by \(\widetilde z\) the image of the complex number \(z\) on the sphere of Riemann \(S\) under the standard map. Let \(D(r)\) be the disc of the radius \(r\) centered at zero and let \(W\) be a meromorphic function. Denote by \[ A(r)= {1\over \pi} \iint_{D(r)} {\bigl| W'(z) \bigr|^2 \over\bigl(1+\bigl| W(z) \bigr|^2 \bigr)^2} dxdy, \quad A\bigl( W(G)\bigr) =\iint_G {\bigl| W'(z) \bigr|^2 \over\bigl( 1+\bigl| W(z) \bigr|^2 \bigr)^2} dxdy. \] The author, in particular, proves the following: Let \(W\) be a meromorphic function and let \(\varphi\) be an increasing function with \(\varphi^{35} (r)<A(r)\). Then there exist a set \(E\subset (1,\infty)\) of a finite logarithmic measure and a family of domains \(E_0(r)\), \(E_1(r), \dots, E_{\Phi^c} (r)\), \(r\in (1,\infty)\) such that: (1) \(D(r)= \bigcup^{\Phi^{c(r)}}_{i=0} E_i(r)\), (2) the function \(W\) is univalent in a domain \(G_i\), \(\overline {E_i(r)} \subset G_i \), \(i=1,2, \dots, \Phi^c(r)\), (3) \(\overline {\widetilde W(E_i(r))} =S- s_i(r)\), \(s_i(r)= \bigcup^{k_i}_{j=1} \Delta^i_j (r)\) where \(\Delta^i_j(r)\) is a simple connected domain, \(\rho (\Delta^i_j(r)) <{1\over \varphi (r)}\) where \(\rho\) is spherical diameter, (4) \(\varlimsup {1 \over \Phi^c(r)} \sum^{\Phi^c (r)}_{i=1} k_i\leq 4\) \((r\to \infty,\;r\overline \in E)\), (5) \(d(E_i(r)) \leq\varphi^7 (r)A^{-1/2} (r)\) where \(d\) is a diameter, \(i\geq 1\), (6) \(\Phi^c (r)A^{-1}(r) \to 1\) \((r \to \infty,\;r\overline\in E)\), (7) \(A(W(E_0(r))) A^{-1} (r)\to 0\) \((r\to\infty,\;r\overline \in E)\). Moreover the author gives estimates of values \({| W'(z) |\over 1+ | W(z) |^2} \), \(\sigma (E_i(r))\), \({\widetilde d(W(z_1), W(z_2)) \over| z_1- z_1|}\) where \(z,z_1\), \(z_2 \in E_i(r)\), \(i\geq 1\), \(\sigma\) is an area, \(\widetilde d\) is a spherical distance. The investigation is close to the Nevanlinna theory and the Ahlfors theory. This question is discussed by the author. Here he unifies and strenghtens the main results of the author's papers (1986), (1991).