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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.

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A better reference on Valiron is

George Valiron, Fonctions analytiques, Presses universitaires de France, 1954, MR0061658, Zbl 0055.06702.

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):

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  • $\begingroup$ Thank you very much for the interesting reference: it is particularly interesting to me the fact that the smallest concave majorant is a generalization (I guess perhaps with an infinity of sides) of Newton's polygon. I've searched for a digital copy of it, but it seem unavailable: I'll try to find a paper copy and have a look at it. $\endgroup$ Jun 11, 2022 at 18:14
  • $\begingroup$ Thank you again for listing Valiron's two original works: I think there's some material to look at. I hope you do not mind for having edited your answer by adding some formatting and linking. $\endgroup$ Jun 15, 2022 at 9:47

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