On a problem of Bloch and Nevanlinna

In [2, p. 138] the question is raised (and attributed to Bloch) whether there exists a bounded function, analytic in the unit circle, whose derivative is not of bounded characteristic. Frostman [1, p. 181] has answered the question affirmatively by constructing a Blaschke product whose derivative is of unbounded characteristic; this product is of course not continuous on the boundary of the unit circle. The theorem of the present note furnishes an example of an absolutely convergent power series (with Hadamard gaps) whose derivative is not of bounded characteristic; this follows from the fact that every function of bounded characteristic has finite radial limits along almost all radii. It also gives a simple example of an analytic function which tends to infinity along almost all radii.