Concentration at a radius for Hardy class functions

In this paper we establish the fundamental properties of concentration at a radius for functions in the classical Hardy space on the unit disk. For f (z) which is not identically zero and given r, 0 < r <1, the concentration is defined via the ratio of the norm of f (rz) to the norm of f (z). Using the Mahler measure of f (z), we obtain information on the distribution of the zeros of f (z) in terms of the concentration ratio. In the last section of the paper, we examine the sharpness of concentration estimates for the Blaschke factor. © 2006 Elsevier Inc. All rights reserved.