# On the function $n \mapsto |a_n|^{\frac 1n}$ for a given power series $\sum_{n} a_n z^n$

I am currently doing research involving power series on the unit disk in $$\Bbb C$$: precisely I am studying the properties of converging power series of a standard form $$f(z)= \sum_{n=0}^\infty a_n z^n \label{1}\tag{1}$$ where (assuming conventionally for the discourse that $$0\notin\Bbb N$$)

1. $$\limsup_{\substack{n\in\Bbb N \\n\to\infty}} |a_n|^{\frac 1n}=1$$, and
2. $$z\in \Bbb D=\{z\in\Bbb C : |z|<1\}$$.

From the basic properties of $$\limsup$$, condition 1 implies the boundedness of the set $$\big\{|a_n|^{\frac 1n}\big\}_{n\in\Bbb N}$$ , i.e. it implies the existence of $$\sup_{\substack{n\in\Bbb N \\n\to\infty}} |a_n|^{\frac 1n}$$, and this has some important consequences on the structure of \eqref{1}, namely an estimate of the size of zero free regions, the size of the minimal univalence radius etc., described in an old paper by Milos Kössler .

The question

The paper  focuses on what a finite value of $$\sup_{\substack{n\in\Bbb N \\n\to\infty}} |a_n|^{\frac 1n}$$ implies: does there exist other studies trying to analyze the implications of a given behavior for $$n \mapsto |a_n|^{\frac 1n}$$, $$n\in \Bbb N$$ or for the related $$n \mapsto \sup_{k>n}|a_k|^{\frac 1k}$$?

References

 Milos Kössler, "O významu čísla $$\sup |a_n|^{\frac 1n}$$ v teorii mocninných řad (The signification of the number $$\sup |a_n|^{\frac 1n}$$ in the theory of power series)", (Czech, English summary), Časopis Pro Pěstování Matematiky a Fysiky 74, No. 1, 47-53 (1949), MR0034833, Zbl 0033.26504