A Problem Concerning the Zeros of a Certain Kind of Holomorphic Function in the Unit Disk

To Hehzut Hasse on his 6~. birthday Let f(a) be a holomorphic. function in the open unit disk D in the complex plane. (b) given any 8 > 0, there exists an n, = no(&) suc,h that, for every IL > rtO, J, lies in the region 1-E < 1 z 1 < 1. Set p!c, = p$ I f(z) I If lim pu, = 00, then we call f, for brevity, an annular function.-CO As is evident from this definition, an annular fun&ion f is not identic.ally constant, and K, the unit circle, is its natural boundary. Furt.hermore, according to Kierst and Szpilrajn ([5], p. 291), every holomorphic function in D has at least one asymptotic value, and an annular function evidently can have only CYI as an asymptotic value; therefore A(f), the set of asymptotic values of f, cont,ains 00 as its sole element. It is known that annular functions exist; we shall refer to examples later. If f is an annular function, denote by Z(f) the set of zeros of f. It follows from a theorem of Collingwood and Cart,wright (131, p. 212, Theorem 9, (ii)), that Z(f) is an infinite set of points in D. Let Z' (/) be the set of limit points of 2 (f). Then clearly Z'(f) s K. We shall be concerned in this article with the following Problem: If f is an anndar function, does 2' (f) = K ? It is known t,hat there exist annular functions for which Z'= K. A function of Koenigs was shown by Fatou ([4], p. 272) t.o be of this nature, and annular functions were constructed by Wolff (see [12]) as well as by Bagemihl, Erdijs, and Seidel (in [1]) in such a way that Z' = K. For each of these functions, every point of K is the end point of an asymptotic path of f. The following theorem enables us to infer that 2' = K for other known annular functions. Theorem 1. Let f be an annular function. Suppose thnt there exists an everywhere dense subset E of K such that every point of E is the end point of an. asymptotic path of f.