1. SeeO. Blumenthal, Sur le mode de croissance des fonctions entières,Bull. Soc. Math. France,35 (1907), pp. 213–232.
2. This, and the further inequalities containingM′(r) have a meaning only for such values ofr whereM′(r) exists.
3. See e. g.W. K. Hayman, A characterization of the maximum modulus of functions regular at the origin,Journal d'Analyse Math.,1 (1951), pp. 135–154-although in a little different form.
4. Inequality (3) is much stronger than the result ofS. M. Shah: $$\overline {\mathop {\lim }\limits_{r = \infty } } \frac{{\log \left( {r\frac{{M_1 (r)}}{{M(r)}}} \right)}}{{\log r{\text{ }}}} = \varrho .$$
5. See the dissertation of the author, „Egészfüggvények maximum-modulus függvényével kapcsolatos vizsgálatok” (Library of the Hungarian Academy of Sciences).