Geodesic
On a Question of Seidel Concerning Holomorphic Functions Bounded on a Spiral
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1255-1262

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

Let S be a spiral contained in D = {|z| < 1} such that S tends to C = {|z| = 1}. For the sake of brevity, by “f is bounded on S” we shall mean that f is holomorphic in D, unbounded, and bounded on S. The existence of such functions was first discussed by Valiron (9; 10); see also (1; 3; 8). Valiron also proved that any function that is “bounded on a spiral” must have the asymptotic value ∞ (10, p. 432). Functions that are bounded on a spiral may also have finite asymptotic values (1, p. 1254). In view of the above, Seidel has raised the question (oral communication): “Does there exist a function bounded on a spiral that has only the asymptotic value ∞?”. The following theorem answers this question affirmatively. 
Barth, K. F.; Schneider, W. J. On a Question of Seidel Concerning Holomorphic Functions Bounded on a Spiral. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1255-1262. doi: 10.4153/CJM-1969-138-8
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