Asymptotic behaviour of pth means of analytic and subharmonic functions in the unit disc and angular distribution of zeros

Abstract

We propose a new approach for studying asymptotic behaviour of pth means of the logarithmic potential and classes of analytic and subharmonic functions in the unit disc. In particular, we generalize a criterion due to G. MacLane and L. Rubel of boundedness of the L2-norm of log |B|, where B is a Blaschke product, in several directions. We describe growth and decrease of pth means, p ∈ (1,∞), for nonpositive subharmonic functions in the unit disc. As a consequence, we obtain a complete description of the asymptotic behaviour of pth logarithmic means of bounded analytic functions in the unit disc in terms of its zeros and the boundary measure. We also prove sharp upper estimates of pth means of analytic and subharmonic functions of finite order in the unit disc.