Karl F. Barth

Extensions of a Theorem of Valiron

A classical theorem of Valiron states that a function which is holomorphic in the unit disk, unbounded, and bounded on a spiral that accumulates at all points of the unit circle, has asymptotic value infinity. This property, and various other properties of such functions, are shown to hold for more general classes of functions which are bounded on a subset of the disk that has a suitably large set of nontangential limit points on the unit circle.

On a Problem of MacLane Concerning Arc Tracts

G. R. MacLane posed the question of whether a locally univalent function in the MacLane class A can have an arc tract. We show that the behaviour of any such example must be extremely irregular. We also indicate one possible approach to constructing an example of such a function, which relates MacLane’s question to the ‘type problem’ for certain Riemann surfaces.

Asymptotic values of strongly normal functions

AbstractLetf be meromorphic in the open unit discD and strongly normal; that is,

Exceptional values and the MacLane class

(1 - |z|^2 )f^\# (z) \to 0as|z| \to 1,

The MacLane class and the Eremenko–Lyubich class

Wheref# denotes the spherical derivative off. We prove results about the existence of asymptotic values off at points ofC=∂D. For example,f has asymptotic values at an uncountably dense subset ofC, and the asymptotic values off form a set of positive linear measure.