This question is based on [this Math.SE answer](https://math.stackexchange.com/questions/4182260/on-the-function-n-mapsto-a-n-frac-1n-for-a-given-power-series-sum-n/4445420#4445420), so let's recall a few concepts dealt with there. If $\{a\_n\}\_{n\in\Bbb N}$ is the sequence of coefficients of a power series $\sum\_{n=0}^\infty a\_nz^n$ with unit convergence radius, then
$$
\limsup\_{n\to+\infty} \ln\lvert a\_n\rvert/n=0\iff \limsup\_{n\to+\infty} \lvert a\_n\rvert^{1\over n}=1.
$$
Shmuel Agmon was able to prove (see [1], chapter 1, §§ 1.1 and 1.2 pp. 264-265, but for a clearer exposition [2], §3 p. 497) that under these assumptions there exists a unique *smallest concave majorant* or *envelope* $\{C(n)\triangleq c\_n\}\_{n>0}$ of the sequence $\{\ln\lvert a\_n\rvert\}\_{n>0}$ (see the references or the cited answer for relevant definitions). In [1], chapter 1, §1.2 p. 265 he states that the function $C(x)$ was introduced by Georges Valiron for the study of entire functions.


**Question**. I wasn't able to locate a precise reference, so does anyone know where Valiron introduces the function $C(x)$ ($C(n)=C(x)|\_{\Bbb N\setminus\{0\}})$?


**Note**. I had a quick look to Valiron's monograph [3] but I was not able to find a definition.


**References**


[1] Shmuel Agmon, "[Sur les séries de Dirichlet](https://doi.org/10.24033/asens.971)" (French), Annales Scientifiques de l’École Normale Supérieure, Troisième (III) Série 66, 263-310 (1949), [MR0033352](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0033352), [Zbl 0034.34602](https://www.zbmath.org/?q=an%3A0034.34602).


[2] Shmuel Agmon, "[Functions of exponential type in an angle and singularities of Taylor series](https://doi.org/10.2307/1990611)" (English), Transactions of the American Mathematical Society 70, 492-508 (1951), [MR0041222](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0041222), [Zbl 0045.34902](https://www.zbmath.org/?q=an%3A0045.34902).


[3] Georges Valiron, *[Lectures on the general Theory of integral functions](https://archive.org/details/lecturesonthegen033156mbp/mode/2up)*, translated by E. F. Collingwood, with a preface by W. H. Young (English) Cambridge: Deighton, Bell and Co., pp. XII+208 (1923), [JFM 50.0254.01](https://www.zbmath.org/?q=an%3A50.0254.01).



Reference(s) on the smallest concave majorant for the sequence of coefficients of a given power series?
A better reference on Valiron is


George Valiron, *Fonctions analytiques*, Presses universitaires de France, 1954, [MR0061658](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0061658), [Zbl 0055.06702](https://zbmath.org/?q=an%3A0055.06702).


This smallest concave majorant is nothing but the Newton polygon generalized to power series. Valiron also calls it Hadamard's polygon.
He has a picture of it in Section 67.


Valiron studies it in the case of entire functions ($\log|a\_n|/n\to-\infty$), but also mentions the case of finite radius of convergence.


Its existence is almost evident: since $\log|a\_n|/n$ is bounded from above, there is a constant majorant. So the class of concave majorants is non-empty, therefore there is a unique smallest concave majorant, since the minimum of any set of concave functions bounded from below is concave.


**Edit**. Valiron refers to his 2 papers in the *Annales scientifiques de l'École Normale Supérieure* (Série 3):


* "[Les théorèmes généraux de M. Borel dans la théorie des fonctions entières](https://doi.org/10.24033/asens.724)", 37, 219-253 (1920), [JFM 47.0301.01](https://zbmath.org/?q=an%3A48.1218.01), and
* "[Recherches sur le théorème de M. Picard](https://doi.org/10.24033/asens.736)", 38, 389-429 (1921), [JFM 48.1218.01](https://zbmath.org/?q=an%3A48.1218.01).


