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Integrals and derivatives of functions in MacLane's class and of normal functions†

Published online by Cambridge University Press:  24 October 2008

K. F. Barth  and
W. J. Schneider
K. F. Barth
Affiliation:
Syracuse University, Syracuse, N. Y.
W. J. Schneider
Affiliation:
Syracuse University, Syracuse, N. Y.
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Extract

If f is meromorphic in D = {,|z| < 1}, we say that f has the asymptotic value a at ζ, ζ ∈ C(={|z| = 1}), if there exists an arc Γ such that Γ ends at ζ, (Γ − {ζ}) ⊂ D, and f has the limit a as |z| → 1 on Γ. The class (M) originally defined by G. R. MacLane ((8), p. 8), consists of those functions that are non-constant and holomorphic (meromorphic) in D and that have asymptotic values at a dense subset of C. In (8), p. 36, MacLane has shown the following three conditions are sufficient for a non-constant holomorphic function f to be in :

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 71 , Issue 1 , January 1972 , pp. 111 - 121
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

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