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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 377.30023

**Autor: ** Bonar, D.D.; Carroll, F.W.; Erdös, Paul

**Title: ** Strongly annular functions with small coefficients, and related results. (In English)

**Source: ** Proc. Am. Math. Soc. 67, 129-132 (1977).

**Review: ** An analytic function f(z) in the unit disc D = **{**z: |z| < 1**}** is called an annular function if there exists a sequence of Jordan curves **{**J_{n}**}** in D such that the origin is in the interior of J_{n} for each n and **lim**_{n ––> oo}**max****{**|f(z)|: z in J_{n}**}** = oo. If, in addition, the curves J_{n} are all circles with center at the origin, then the function f(z) is said to be strongly annular. The authors construct an example of a strongly annular function f(z) = **sum**_{n = 0}^{oo}a_{n}z^{n} such that **lim**_{n ––> oo}a_{n} = 0. The construction is very short and elementary. Additional examples of annular functions are presented in which various length and distance apart conditions are placed on the curves J_{n}. These additional examples involve approximation techniques.

**Reviewer: ** P.Lappan

**Classif.: ** * 30D40 Cluster sets, etc.

30B10 Power series (one complex variable)

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