Geodesic
On the loci |f(z)| = R, f(z) Entire
Glasgow mathematical journal, Tome 10 (1969) no. 2, pp. 94-102

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

The following result is found quite widely. Suppose f(z) is a non-constant entire function such that |f(z)| = 1 along |z| = 1. Then, f (z) has form czm, |c| = 1, m ≧ 1. See Ahlfors [1, p. 172, exercise 3], Dienes [4, p. 172, exercise 23], Hille [6, p. 317, exercise 2]. It is natural to inquire about a generalization of this result. 
Hejhal, Dennis A. On the loci |f(z)| = R, f(z) Entire. Glasgow mathematical journal, Tome 10 (1969) no. 2, pp. 94-102. doi: 10.1017/S0017089500000628
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