$Q_{\alpha}$-Normal Families and entire functions

TL;DR

This paper constructs specific entire functions for each countable ordinal to demonstrate their associated families are exactly $Q_eta$-normal in the unit disk, advancing the understanding of normal families in complex analysis.

Contribution

It introduces a method to construct entire functions corresponding to each countable ordinal with precisely defined $Q_eta$-normality properties.

Findings

For each countable ordinal, an entire function with a $Q_eta$-normal family is constructed.

The family $ig\{f(nz): n ext{ in } atsig\}$ exhibits exact $Q_eta$-normality.

The work links ordinal theory with complex normality concepts.

Abstract

For every countable ordinal number $\alpha$ we construct an entire function $f=f_\alpha$ such that the family $\left\{f(nz):n\in\mathbb{N}\right\}$ is exactly $Q_\alpha$-normal in the unit disk.