On the boundary behavior of functions for which the Riemann image has finite spherical area (Q1074738)

In this paper there is proved the following result and a number of applications of it are given. Let f be meromorphic in the half-strip \(S: a<x<b\), \(y>0\) and let the Riemann image of S by f have finite spherical area. Let \(\sigma\) denote the boundary point of S at the point at infinity. Let T be a closed subset of S such that for fixed \(L>0\) every rectangle \(a<x<b\), \(Y\leq y\leq Y+L\) (Y\(\geq 0)\) contains a subcontinuum of T of diameter \(\geq \delta\), \(\delta >0\). Let U be a subset of \(a'\leq x\leq b'\), \(y>0\) with \(a<a'<b'<b\). Then for the cluster sets of f at \(\sigma\) on T and U we have \(C(f,T,\sigma)\supset C(f,U,\sigma)\).