Normality and boundary behavior of arbitrary and meromorphic functions along simple curves and applications

Abstract

We establish the theorems that give necessary and sufficient conditions for an arbitrary function defined in the unit disk of complex plane in order to has boundary values along classes of equivalencies of simple curves. Our results generalize the well--known theorems on asymptotic and angular boundary behavior of meromorphic functions (Lindolf, Lehto--Virtanen, and Seidel--Walsh type theorems). The results are applied to the study of boundary behavior of meromorphic functions along curves using $P-$sequences, as well as in the proof of the uniqueness theorem similar to Saginjan's one. Constructed examples of functions show that the results cannot be improved.