Reduced moduli of open sets in the theory of analytic functions
Let \(B\) be an open set of the extended complex plane \(\overline C\); \(z_i\), \(i= 1,\dots, m\), be different points of the set \(B\); \(\delta_i\), \(i=1,\dots,m\), be arbitrary real numbers not all zero; \(\mu_i\) and \(\nu_i\), \(i= 1,\dots, m\), be arbitrary positive numbers. The author uses the follwoing notations \(Z= \{z_i\}\), \(\Delta= \{\delta_i\}\), \(\Psi= \{\psi_i\}\), \(\psi_i= \psi(r)\equiv \mu_i r^{\nu_i}\), \(i= 1,\dots, m\); for \(r>0\), \(E(z_0, r)= \{z:|z- z_0|\leq r\}\) if the point \(z_0\) is finite and \(E(z_0, r)= \{z:|z|>{1\over r}\}\) when \(z_0=\infty\). For a sufficiently small \(\tau>0\), a generalized condenser is defined as the ordered set \[ C(r; B,Z,\Delta, \Psi)= \{E_0, E(z_1, \psi_1(r)),\dots, E(z_m,\psi_m(z))\} \] with present values \(0\), \(\delta_1,\dots,\delta_m\), respectively. The modulus of the condenser \(C(r; B,Z,\Delta,\Psi)\) is the value \[ |C(r; B,Z,\Delta,\Psi)|= \sup\Biggl\{1/ \iint_C|\nabla \nu|^2 dx dy\Biggr\}, \] where the upper bound is taken over all real-valued functions \(\nu(z)\) that are continuous in \(\overline C\), satisfying the Lipshitz condition in some neighborhood of each finite point, vanish in the neighborhood of the set \(E_0\) and take on the values \(\nu(z)= \delta_i\) on \(E(z_i,\psi_i(r))\) for \(i= 1,\dots, m\), \((1/0)= +\infty\). The author uses the term reduced modulus of the set \(B\) with respect to the sets \(Z\), \(\Delta\) and \(\Psi\) for the limit \[ M(B, Z,\Delta,\Psi)= \lim_{r\to 0} (C|r;B,Z, \Delta,\Psi)|+ \nu\log r/2\pi), \] where \(\nu= (\sum\delta^2_i, \nu^{-1}_i)^{-1}\). In the paper one proves a theorem on the existence of the reduced modulus and there are given some applications to the theory of analytic functions.
scientific article; zbMATH DE number 1557560