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Bulletin of the London Mathematical Society (1)

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EXTENSIONS OF A THEOREM OF VALIRON
KARL F. BARTH, PHILIP J. RIPPON
Journal:

Bulletin of the London Mathematical Society / Volume 38 / Issue 5 / October 2006

Published online by Cambridge University Press:

20 September 2006, pp. 815-824

Print publication:

October 2006
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A classical theorem of Valiron states that a function which is holomorphic in the unit disk, unbounded, and bounded on a spiral that accumulates at all points of the unit circle, has asymptotic value $\infty$. This property, and various other properties of such functions, are shown to hold for more general classes of functions which are bounded on a subset of the disk that has a suitably large set of nontangential limit points on the unit circle.