Geodesic
The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution
Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 803-814

Voir la notice de l'article provenant de la source Cambridge University Press

Let f(z) be a function meromorphic in the unit disc D = (|z| < 1). We consider the maximum modulus and the minimum modulus When no confusion is likely, we shall write M(r) and m(r) in place of M(r,f) and m(r,f).Since every normal holomorphic function belongs to an invariant normal family, a theorem of Hayman [6, Theorem 6.8] yields the following result.THEOREM 1. If f(z) is a normal holomorphic function in the unit disc D, then (1) This means that for normal holomorphic functions, M(r) cannot grow too rapidly. The main result of this paper (Theorem 5, also due to Hayman, but unpublished) is that a similar situation holds for normal meromorphic functions. 
Gauthier, Paul. The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution. Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 803-814. doi: 10.4153/CJM-1970-090-7
@article{10_4153_CJM_1970_090_7,
     author = {Gauthier, Paul},
     title = {The {Maximum} {Modulus} of {Normal} {Meromorphic} {Functions} and {Applications} to {Value} {Distribution}},
     journal = {Canadian journal of mathematics},
     pages = {803--814},
     year = {1970},
     volume = {22},
     number = {4},
     doi = {10.4153/CJM-1970-090-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-090-7/}
}
TY  - JOUR
AU  - Gauthier, Paul
TI  - The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution
JO  - Canadian journal of mathematics
PY  - 1970
SP  - 803
EP  - 814
VL  - 22
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-090-7/
DO  - 10.4153/CJM-1970-090-7
ID  - 10_4153_CJM_1970_090_7
ER  - 
%0 Journal Article
%A Gauthier, Paul
%T The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution
%J Canadian journal of mathematics
%D 1970
%P 803-814
%V 22
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-090-7/
%R 10.4153/CJM-1970-090-7
%F 10_4153_CJM_1970_090_7

[1] 1. Gauthier, P. M., A criterion for normalcy, Nagoya Math. J. 32 (1968), 277–282. Google Scholar

[2] 2. Gauthier, P. M., Cercles de remplissage and asymptotic behaviour, Can. J. Math. 21 (1969), 447–455. Google Scholar

[3] 3. Gauthier, P. M., Cercles de remplissage and asymptotic behaviour along circuitous paths, Can. J. Math. 22 (1970), 389–393. Google Scholar

[4] 4. Gavrilov, V. I., On a theorem of A. L. Shaginjan, Vestnik Moskov. Univ. Ser I Mat. Meh., no. 2 (1966), 3–10. (Russian) Google Scholar

[5] 5. Hayman, W. K., On Nevanlinna1 s second theorem and extensions, Rend. Circ. Mat. Palermo (2) 2 (1953), 346–392. Google Scholar

[5] 5. Hayman, W. K., Meromorphic functions (Clarendon Press, Oxford, 1964). Google Scholar

[7] 7. Heins, M. H., The minimum modulus of a bounded analytic function, Duke Math. J. 14 (1947), 179–215. Google Scholar

[8] 8. Hille, E., Analytic function theory, Vol. 2 (Ginn, Boston, 1962). Google Scholar

[9] 9. Lange, L. H., Sur les cercles de remplissage non-euclidiens, Ann. Sci. Ecole Norm. Sup (3) 77 (1960), 257–280. Google Scholar

[10] 10. Rung, D. C., Behavior of holomorphic functions in the unit disc on arcs of positive hyperbolic diameter, J. Math. Kyoto Univ. 8 (1968), 417–464. Google Scholar

[11] 11. Schnitzer, F. and Seidel, W., On the rate with which a holomorphic function in a disk can tend radially to zero, Proc. Nat. Acad. Sci. U.S.A. 67 (1967), 876–877. Google Scholar

[12] 12. Shaginjan, A. L., A fundamental inequality in the theory of functions and its applications, Izv. Akad. Nauk Armjan. SSR Ser. Fiz.-Mat. Nauk 12 (1959), no. 1, 3–25. (Russian. Armenian Summary) Google Scholar

[13] 13. Walsh, J. L., Interpolation and approximation by rational functions in the complex domain, 3rd éd. (Amer. Math. Soc, Providence, Rhode Island, 1960). Google Scholar

Cité par Sources :