Mean growth and smoothness of analytic functions

Abstract:

Let ${G_\alpha }$ denote the class of functions $f(z)$ analytic in the unit disk such that \[ \int _0^1 {{{(1 - r)}^{ - \alpha }}{M_\infty }(f’,r)} dr < \infty ,\] for some $\alpha (0 < \alpha < 1)$. A characterization of ${G_\alpha }$ is given in terms of moduli of continuity and an application is given to Riesz factorization of functions in ${G_\alpha }$.