VOLUME INTEGRAL MEANS OF HOLOMORPHIC FUNCTIONS

The classical integral means of a holomorphic function f in the unit disk are defined by [1/2 pi integral(2 pi)(0) vertical bar f(re(i theta))vertical bar(p) d theta](1/p), 0 <= r < 1. These integral mea is play an important role in modern complex analysis. In this note we consider integral means of holomorphic functions in the unit ball B-n in C-n with respect to weighted volume measures, M-p,M-alpha(f, r) = [1/v(alpha)(rB(n)) integral(rBn) vertical bar f(z)vertical bar(p) dv(alpha)(z)](1/p), 0 <= r < 1, where alpha is real, dv(alpha)(z) = (1 - vertical bar z vertical bar(2))(alpha) dv(z), and dv is volume measure on B-n. We show that M-p,M-alpha(f, r) increases with r strictly unless f is a constant, but in contrast with the classical case, log M-p,M-alpha(f, r) is not always convex in log r. As an application, we show that if alpha <= -1, M-p,M-alpha(f, r) is bounded in r if and only if f belongs to the Hardy space H-P, while if a > -1, M-p,M-alpha(f, r) is bounded in r if and only if f is in the weighted Bergman space A(alpha)(p).