02-4   Berichtsreihe des Mathematischen Seminars der Universität Kiel

## On proper analytic maps with one critical point

Let U \subset C be a domain containing
{ z \in \C : \re z > \sigma, ¦ \arg z ¦ < \eta }
for some \sigma,\eta>0 and let \$f:U ->U be a proper holomorphic map satisfying f(z) = z+1+a/z+o\left(1/¦z¦\right) as z\to\infty, ¦\arg z¦<\eta, with a \in \C. We show that if U contains only one critical point of f, and this critical point is simple, then \re a \geq \frac14. This slightly generalizes a previous result concerning critical points in Leau domains. We also show that the condition \re a \geq \frac14 is sharp.

Mathematics Subject Classification (1991): Primary 37F10; Secondary 30D05, 37F45

Bibliographical note: erscheint in einem Tagungsbericht: Reihe "Contemp. Math." der AMS