This question is based on this Math.SE answer, 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" (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] Georges Valiron, *Lectures on the general Theory of integral functions*, 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.