International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 3, Pages 369-413

The basic properties of Bloch functions

Joseph A. Cima

Mathematics Department, The University of North Carolina at Chapel Hill, Chapel Hill 27514, North Carolina, USA

Received 2 February 1979

Copyright © 1979 Joseph A. Cima. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A Bloch function f(z) is an analytic function on the unit disc 𝔻 whose derivative grows no faster than a constant times the reciprocal of the distance from z to 𝔻. We reprove here the basic analytic facts concerning Bloch functions. We establish the Banach space structure and collect facts concerning the geometry of the space. We indicate duality relationships, and known isomorphic correspondences are given. We give a rather complete list of references for further study in the case of several variables.