On the Bloch-Nevanlinna problem

Abstract:

In 1929, A. Bloch and R. Nevanlinna [5] posed the problem of whether the derivative of a function of bounded characteristic is also of bounded characteristic. Since then counter-examples have been constructed by Frostman, Rudin, Lohwater, Piranian, Hayman, and Duren ([1], [2], [3], [4], [6], [7]). In this short note we prove that in the Banach space of absolutely convergent power series the subset consisting of members whose derivatives have no finite radial limit in any direction is dense. Since the radial limit of a function of bounded characteristic exists in almost every direction, the result disproves the conjecture in a rather emphatic form. Similar results are established for the space of functions which are analytic in the unit disc and continuous in the closed disc, and also for the space of absolutely convergent Fourier series. The existential proof given here seems to be easier than any of the known constructive proofs, and actually establishes more.