On the Boundary Behavior of Holomorphic Functions in the Unit Disk

Abstract

Let f(z) be a holomorphic function defined in the unit disk |z|<1, which we shall denote by D. Let Σ be a subset of D, whose closure has at least one point in common with C, the circumference of the unit disk. The set of all values a such that the equation f(z) = a has infinitely many solutions in Σ is called the range of f(z) in Σ, and is denoted by R(f, Σ). Let τ be a point of C, and let {zn ) be a sequence of points in D with the properties: . The non-Euclidean (hyperbolic) distance ρ(zn, zn+1 ) between two points zn and z n+1 of the sequence is defined to be equal to (cf.[3], Ch. II).