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Sums and products of functions in the MacLane class A
Published online by Cambridge University Press: 26 February 2010
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Let be the class of non-constant functions f(z), holomorphic in |z| < 1, which have asymptotic values at a dense set of points on |z| = 1. MacLane [4; p. 18] asked whether the sum and product of two functions in had to be either constant or in . Recently Barth and Ryan [2] have shown that this was not necessarily so; in this note we will demonstrate, in a totally elementary way, why not. Our principal result is:
Theorem 1. Any non-constant function R(z), holomorphic in |z| < 1, may be represented both as a sum and as a product of pairs of functions in.
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