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2 results

  • On the asymptotic boundary behavior of functions analytic in the unit disk

  • James R. Choike
  • Journal:
    Nagoya Mathematical Journal / Volume 67 / August 1977
    Published online by Cambridge University Press:
    22 January 2016, pp. 1-13
    Print publication:
    August 1977
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  • In [8] a necessary and sufficient condition was given for determining the equivalence of two asymptotic boundary paths for an analytic function w = f(p) on a Riemann surface F. In this paper we give a necessary and sufficient condition for determining the nonequivalence of two asymptotic boundary paths for f(z) analytic in |z| < R, 0 < R ≤ + ∞. We shall, also, illustrate some applications of the main result and examine a class of functions introduced by Valiron.

  • On the Distribution of Values of Functions in the Unit Disk

  • James R. Choike
  • Journal:
    Nagoya Mathematical Journal / Volume 49 / March 1973
    Published online by Cambridge University Press:
    22 January 2016, pp. 77-89
    Print publication:
    March 1973
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  • Let f(z) be a function analytic and bounded, |f(z)| < 1, in |z| < 1. Then, by Fatou’s theorem the radial limit f*(eiθ ) = limr→1f(reiθ ) exists almost everywhere on |z| = 1. Seidel [8, p. 208] and Calderón, González- Domínguez, and Zygmund [1] (see also [9, pp. 281-282]) proved the following: if f*(eiθ ) is of modulus 1 almost everywhere on an arc a < θ < b of |z| = 1, then either f(z) is analytically continuable across this arc or the values f*(eiθ ), a < d < b, cover the circumference |w| = 1 infinitely many times.