Phragmén-Lindelöf-principle

Question

I'm currently working on my bachelor thesis and for that I need to learn more about the Phragmén-Lindelöf principle. I found the original article written by the two mathematicians, but it is in french (Here). So I'm trying to translate it, but I have a few problems. What does ist mean, that $\vert f(x)\vert$ for a function $f$ has to be uniform ("Supposons que le module $\vert f(x)\vert$ est uniforme dans le domaine $T$" (p.1))? Is a regular function just a holomorphic function or something else?

Answer 1

I'll sum up what I said in my comments and make it into an answer, although if someone with actual knowledge in old French mathematical vocabulary wants to correct or complete what I said I would be more than happy to edit my answer.

From looking at some now century-old French papers about holomorphic functions (which seemed to not always be named so at the time, instead sometimes being called "régulière", aka regular, in papers like the one you linked, or "monogène" in e.g. the Leçons sur les fonctions monogènes uniformes d'une variable complexe by Émile Borel (name by Cauchy it seems)), it would seem like "uniforme" in this context had the meaning of single-valued, the modern French term being "univaluée", while "multiforme" would have the meaning of multi-valued, "multivaluée". I'll refer to this paper by Paul Montel, Sur les familles de fonctions analytiques qui admettent des valeurs exceptionnelles dans un domaine, Annales scientifiques de l’É.N.S. 3e série, tome 29 (1912), p. 487-535, for a use of both uniforme and multiforme right in the Introduction portion, but also more importantly for a paragraph at pages $494-495$ (pages $9$-$10$ of the file) where a function $\varphi$ is justified to be uniforme in a few sentences that should prove exactly singlevaluedness (probably not a real word but oh well), which I'll translate to the best of my ability:

"(...) $7$. Let's now consider the case where the functions $f(x)$ [they're holomorphic functions defined on a connected domain $D$] don't take any of the values represented by the points of an open curve $\Gamma$ of the $X$-plane joining the point $a$ to the point $b$. In other words, the domain $\Delta$ [$f(D)$ I assume] is not traversed by $\Gamma$. I claim that the functions $f(x)$ form a normal family.
Define $$\varphi : x \mapsto \log\left(\frac{f(x) - b}{f(x) - a}\right)$$ Each function $\varphi$ is "uniforme" in $D$ since, $X = f(x)$ never intersecting $\Gamma$, this point, when $x$ runs along a closed contour, can only go along closed curves which include either both or neither of $a$ and $b$ in their interior. (...)"

This use of uniforme for single-valued is furthermore backed up by one of its entries in the Centre National de Ressources Textuelles et Lexicales, see https://www.cnrtl.fr/definition/uniforme : "II. − MATH., PHYS. A. − Qualifie une grandeur, soit constante dans une région de l'espace (ex: champ uniforme) soit constante au cours du temps (ex: mouvement uniforme) (Mathieu-Kastler 1983). B. − Fonction uniforme dans un domaine. ,,Fonction de la variable complexe f(z) qui, en tout point z de ce domaine, ne peut prendre qu'une seule valeur (Lar. Lang. fr.)." (There is also an entry for "monogène" on there which essentially says that monogène = differentiable).

Finally, it does not seem that there is another meaning when applied to the modulus of a function instead of the function itself, such as it referring to uniform continuity or something else, but rather it would appear that there is (or was? I do not know) simply an interest in possibly multi-valued whose moduli are forced to be single-valued. One example that comes to mind would be the square root function, since both square roots have the same modulus, thus the function "modulus of square roots" is single-valued. See this paper by Leo Sario for an example of a paper studying such classes of functions: On locally meromorphic functions with single-valued moduli.