Barth, K. F. and Rippon, Philip
Extensions of a theorem of Valiron.
Bulletin of the London Mathematical Society, 38(5) pp. 815–824.
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 infinity. 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.
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