On omitted tuples for univalent functions

✍ Scribed by Ragnhild Johanne Rensaa

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 92 KB
• Volume: 105
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(99)00011-4

✦ Synopsis

In the family S of normalized, univalent functions, an omitted point in F ⊂ S is a complex number w0, such that there is at least one function f ∈ F, satisfying f(z) = w0 for all |z|¡1:

Let a set of m distinct complex numbers w1; w2; : : : ; wm all = 0, be given such that 06arg w1¡arg w2¡ • • • ¡arg wm¡2 . The tuple (w1; w2; : : : ; wm) shall be called an omitted tuple for F if there exists at least one f ∈ F such that f(z) = wi, ∀i = 1; 2; : : : ; m and all |z|¡1. In this paper we shall be concerned with the question whether (t w1; t w2; : : : ; t wm), t an arbitrary positive number, is an omitted tuple in S or not, more precisely the number of functions omitting the tuple for di erent values of t. An answer to this question in full generality will not be o ered, but some partial results are given. Moreover, two subfamilies where complete solutions are obtained, are brie y mentioned.