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

Abstract

In this paper, we establish some theorems giving necessary and sufficient conditions for an arbitrary function defined in the unit disc of the complex plane to have boundary values along classes of an equivalence relation over simple curves. Our results generalize the well-known theorems on asymptotic and angular boundary behaviours of meromorphic functions (Lindelölf-, Lehto–Virtanen- and Seidel–Walsh-type theorems). The obtained results are applied to the study of boundary behaviour of meromorphic functions along curves using P-sequences, as well as in the proof of the uniqueness theorem similar to Šaginjan’s one. The constructed examples of functions show that the results cannot be improved.

Keywords:

• Normal functions
• normal families of functions
• angular limits of analytic functions
• simple curves
• the hyperbolic distance
• the Fréchet distance between curves

AMS Subject Classifications:

• Primary 30D40
• Secondary 51K99

Acknowledgements

We are thankful to Professor Romeo Meštrović for very useful comments. We also would like to thank the referee for the careful reading of earlier versions of the manuscript and for his/her comments which improved the quality of the manuscript. Also, we would like to dedicate this work to the memory of Prof. Valerian I. Gavrilov, who passed away during preparing the last version of this article and who communicated some suggestions to us.

Notes

No potential conflict of interest was reported by the authors.

This work is supported by the Competitiveness Program of NRNU MEPhI and dedicated to the memory of Prof. Valerian I. Gavrilov.

Additional information

Funding

This work is supported by the Competitiveness Program of NRNU MEPhI.