Applications of the theory of cluster sets to a class of meromorphic functions

Abstract

Let f (z) be meromorphic and non-rational in the domain |z| < R ≤ ∞, and let a be an arbitrary complex number, which may be infinite. The deficiency δ(a) of the value a is defined bywhere m(r, a), N(r, a) and T(r) are defined as usual (cf. (10), pp. 156 ff.). For the class of functions considered in this paper the characteristic function T(r) is unbounded, and this will be assumed throughout. The upper (or Valiron) deficiency (16) of the value a is denned byfrom which it follows that 0 ≤ δ(a) ≤ Δ(a) ≤ 1. A value a for which Δ(a) > 0 is called exceptional or deficient, and a value for which Δ(a) = 0 is called normal. We shall denote by G[a, σ] the open set of all values z in | z | < R for which | f(z) – a | < σ, where σ is a given positive number; we shall say that a component Gn[a, σ] of G[a, a] is bounded if the closure G¯n[a, σ] is contained in | z | < R, otherwise Gn[a, σ] will be called unbounded. In the case a = ∞, it is natural to define Gn[∞, σ] as the set of all z for which | f(z) | > 1/σ.