On the multiple points of certain meromorphic functions

Abstract

Abstract.We show that if / is transcendental and meromorphic in the plane and T(r, f) = o(logr)2 , then / has infinitely many critical values. This is sharp. Further, we apply a result of Eremenko to show that if / is meromorphic of finite lower order in the plane and N(r, l/ff") = o(T(r, f If)), then f{z) = exp(az + b) or f(z) = (az + b)~" with a and b constants and n a positive integer.

Entire functions with Julia sets of positive measure

“…As we shall see below, we have ( f ) ≥ 1 2 for f ∈ B. (This seems to have been observed first in [4,19]; see also [24,Lemma 3.5].) Thus N ≤ 2 ( f ) in this case.…”

Transcendental meromorphic functions with three singular values

“…Langley [11,12] discovered that there exists a relation between the number of singular values of a meromorphic function f and the growth of the Nevanlinna characteristic T (r, f ). In [12] he proved that all meromorphic functions f with finitely many singular values satisfy lim inf r→∞ T (r, f ) log 2 r > 0.…”

The growth rate of an entire function and the Hausdorff dimension of its Julia set

“…Next we note that, if f ∈ B, then ρ(f ) ≥ 1 2 . This observation seems to have appeared first in [13], [6]; see also [18,Lemma 3.5]. This implies that a function f ∈ B cannot satisfy (1.2) for some q < 1.…”