Volume Integral Means of Holomorphic Functions

The classical integral means of a holomorphic function f in the unit disk are defined by [ 1 2π ∫ 2π 0 |f(re)| dθ ]1/p , 0 ≤ r < 1. These integral means play an important role in modern complex analysis. In this note we consider integral means of holomorphic functions in the unit ball Bn in C with respect to weighted volume measures, Mp,α(f, r) = [ 1 vα(rBn) ∫ rBn |f(z)| dvα(z) ]1/p , 0 ≤ r < 1, where α is real, dvα(z) = (1− |z|) dv(z), and dv is volume measure on Bn. We show that Mp,α(f, r) increases with r strictly unless f is a constant, but in contrast with the classical case, logMp,α(f, r) is not always convex in log r. As an application, we show that if α ≤ −1, Mp,α(f, r) is bounded in r if and only if f belongs to the Hardy space H, while if α > −1, Mp,α(f, r) is bounded in r if and only if f is in the weighted Bergman space Aα.