J. Austral. Math. Soc.  72 (2002), 131-136
Growth of functions in cercles de remplissage

P. C. Fenton
  Department of Mathematics
  University of Otago
  Dunedin
  New Zealand
  pfenton@maths.otago.ac.nz
and
John Rossi
  Department of Mathematics
  Virginia Tech
  Blacksburg VA 24060
  USA
  rossi@calvin.math.vt.edu


Abstract
Suppose that  f is meromorphic in the plane, and that there is a sequence $z_n\to\infty$ and a sequence of positive numbers $\epsilon_n \to 0$, such that $\epsilon_n|z_n|f^{\#} (z_n)/\log |z_n | \to \infty$. It is shown that if  f is analytic and non-zero in the closed discs $\Delta_n = \{ z : |z - z_n| \leq \epsilon _n |z_n|\}$,  n = 1, 2, 3, . . . , then, given any positive integer  K, there are arbitrarily large values of  n and there is a point  z in $\Delta_n$ such that $|f(z)| > |z|^K$. Examples are given to show that the hypotheses cannot be relaxed.
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