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

analytic-functionspower seriesreference-request

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 [1].

The question

The paper [1] 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

[1] 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

Best Answer

Edit: I've modified my latest note by including materials on references I got aware thanks to this answer

At the beginning of its mathematical career, Shmuel Agmon wrote several complex analysis papers dealing with the singularities of Taylor series, possibly under the influence of its advisor Szolem Mandelbrot. In one of them (see [1], chapter 1, §§ 1.1 and 1.2 pp. 264-265, but for a clearer exposition [2], §3 p. 497), he considers a class of real valued function $\omega:\Bbb R_{\ge 0}\to\Bbb R$ such that $$ \limsup_{x\to+\infty} \frac{\omega(x)}{x}=0 $$ (and note that $\limsup_{n\to+\infty} \ln|a_n|/n=0\iff \limsup_{n\to+\infty} |a_n|^{1\over n}=1$) he proves that, for each function $\omega$ in this class there exists a function $C(x)$ such that

  1. $C(x)$ is a majorant for $\omega(x)$ in the sense that $C(x)\ge\omega(x)$,
  2. $C(x)$ is concave, i.e. $C(x+h)+C(x-h)-2C(x)\le 0$ for $h>0$ and $x-h\ge 0$,
  3. $C(x)$ is the unique smallest real function having the two preceding properties.

He calls the function $C(x)$ the smallest concave majorant or envelope of $\omega$.
Now it is clear that, by considering the real function $|a_n|:\Bbb N\to \Bbb R_{\ge 0}$, the above results imply that there exists its smallest concave envelope $C(n) =c_n$, and it is such that $$ c_n \ge \ln|a_n|\iff e^{c_n} \ge |a_n| \ge 0. \label{2}\tag{2} $$ Equation \eqref{2} allows already for an estimation of the coefficients of \eqref{1}: however, a more important property of $C(x)$ ([2], §3 p. 497) is known, i.e. that $$ \lim_{x\to+\infty}\frac{C(x)}{x}=0 \iff \lim_{x\to+\infty}\frac{c_n}{n}=0 $$ and this implies that $e^{c_n} = O(k^n)$ for some $k>0$ and $n>N$ as $n\to \infty$. And if, for example, \eqref{1} converges for $z=1$, then \eqref{2} can be used to deduce precise power like decreasing estimates for the its sequence of coefficients $\{ a_n\}_{n\in\Bbb N}$.

Notes

  • I found references [1] and [2] nearly two months ago, while searching for references related to the classical Fatou-Riesz theorem. It was some kind of surprise for me since I acknowledged Agmon as a specialist in PDEs and not in complex analysis: however, it was a good surprise.

  • In [1], chapter 1, §1.2 p. 265 Agmon states that the function $C(x)$ was introduced by Georges Valiron for the study of entire functions: thanks to this answer by Alexandre Eremenko, I had a look at the original references (see the cited answer for bibliographic indications). The construction of $C(x)$ is a generalization to power series of the construction of Newton's polygon for a polynomial: Valiron goes by calling it "Hadamard polygon" as Hadamard seems the first to have used this construction in his famous 1893 paper on the Riemann function ([3], §2-5, pp. 172-175).

References

[1] Shmuel Agmon, "Sur les séries de Dirichlet" (French), Annales Scientifiques de l’École Normale Supérieure, Troisième (III) Série 66, 263-310 (1949), MR0033352, Zbl 0034.34602.

[2] Shmuel Agmon, "Functions of exponential type in an angle and singularities of Taylor series" (English), Transactions of the American Mathematical Society 70, 492-508 (1951), MR0041222, Zbl 0045.34902.

[3] Jacques Hadamard, "Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann"(French), Journal de Mathématiques Pures et Appliquées, Quatrième (4) Série, vol. IX, 171-215 (1893), JFM 25.0698.03.

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