Radially distributed values and normal families

TL;DR

This paper investigates the normality of families of holomorphic functions in the unit disk with zeros and 1-points constrained to specific rays, revealing conditions for normality and analyzing special cases in detail.

Contribution

It establishes the normality of such function families with zeros and 1-points on distinct rays, and provides a detailed study of the case with positive and negative real axes.

Findings

Family is normal in the punctured disk

Normality holds when zeros and 1-points are on distinct rays

Special case with positive and negative real axes analyzed in depth

Abstract

Let $L_0$ and $L_1$ be two distinct rays emanating from the origin and let ${\mathcal F}$ be the family of all functions holomorphic in the unit disk ${\mathbb D}$ for which all zeros lie on $L_0$ while all $1$-points lie on $L_1$. It is shown that ${\mathcal F}$ is normal in ${\mathbb D}\backslash\{0\}$. The case where $L_0$ is the positive real axis and $L_1$ is the negative real axis is studied in more detail.