The boundary behaviour of derivatives of univalent functions

Abstract

We shall be concerned in this paper with the derivatives of a class of functions f(z) which are analytic and univalent in the unit disc D: ]z[ < 1. It is known [2], for example, that the derivative f' (z) of a univalent function can have radial limits only on a set of measure zero on the circle K: ]z[ = 1. [Indeed, this class of univalent functions was constructed to settle a problem of Bloch and Nevanlinna ([-4], p. 138), namely, whether the derivative of a function of bounded characteristic is also of bounded characteristic. Since the derivative of a function in the class of univalent functions described above possesses radial limits on a set on K of measure zero, it is clear that the derivative of a function in this class cannot be of bounded characteristic, so that the Bloch-Nevanlinna problem has a negative answer. For Frostman's original solution, see [2].] However, it has been observed by McMillan that it is a trivial consequence of the Koebe Verzerrungssatz that the derivative f' (z) of any univalent function is a normal analytic function (cf. [1]), so that, by a theorem of MacLane [-3], f'(z) possesses angular limits at a set of points which is dense on K. Now the class of univalent functions which we shall examine in this paper, namely, the class of univalent functions whose derivatives possess radial limit values on a set of measure zero on K, will prove useful in demonstrating certain properties of the class of all univalent functions, as in Theorem 1, where we prove that the derivative of any univalent function possesses radial limits (and hence angular limits) at a non-denumerable set on K. If we compare the derivatives of this special class of univalent functions with the modular function, which is also normal in D, but has the property that the dense set on K at which it possesses angular limits is denumerable, we see that this special class of univalent functions is of independent interest. In particular, Theorem 3 provides a sufficient condition for the existence of radial limit values other than 0 and ~ for the derivative of a function of the class under consideration.