Radially Distributed Values and Normal Families

Abstract

Let L 0 and L 1 be two distinct rays emanating from the origin and let F be the family of all functions holomorphic in the unit disk D for which all zeros lie on L 0 while all 1-points lie on L 1 . It is shown that F is normal in D\{0}. The case where L 0 is the positive real axis and L 1 is the negative real axis is studied in more detail.

Meromorphic Functions with Three Radially Distributed Values

“…Let D(a, r ) denote the closed disk of radius r around a point a. The following result was proved in [3,Thm. 1.3] and [4,Prop.…”

On unavoidable families of meromorphic functions

“…To give an example, we recall that a classic result states that for two distinct rays and emanating from the origin, there is no transcendental entire function for which all zeros lie on and all ones lie on , while any (nonconstant) polynomial having this property is of degree 1 (e.g., [1, 2]). Hence, given two sequences and having no finite limit point and such that has at least two elements, the set is not a zero–one set.…”

“…Hence, if the function g ∈ M(C) avoids f 1 and f 2 , it cannot avoid 2 f1 f2 f1+ f2 , and the family { f 1 , f 2 , 2 f1 f2 f1+ f2 } is unavoidable with respect to M(C). By Proposition 1, the functions f 1 and − f 2 cannot avoid each other, and because they further have no common zeros, there exists (1), we consider the function…”