Growth of functions in cercles de remplissage

Abstract

Suppose that f is meromorphic in the plane, and that there is a sequence z(n) --> infinity and a sequence of positive numbers epsilon(n) --> 0, such that epsilon(n)\z(n)f(#)(z(n))/log\z(n)\ --> infinity. It is shown that if f is analytic and non-zero in the closed discs Delta(n) = {z : \z - z(n)\ less than or equal to 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.