Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of $\mathbf {C}^ n$

Abstract:

In this paper, the following two inequalities are proved: \[ \int _0^1 {{(1 - r)}^{a|\alpha | + b}}M_p^a(r,D^{\alpha } f) dr \leq K\int _0^1 {{(1 - r)}^b}M_p^a(r,f) dr, \int _0^1 {{(1 - r)}^b}M_p^a(r,f) dr \\ \leq K\left \{ \sum \limits _{|\alpha | \leq m - 1} |({D^\alpha }f)(0)|^a + \sum \limits _{|\alpha | = m} \int _0^1 {(1 - r)}^{am + b}M_p^a(r,D^{\alpha }f) dr \right \} \] where $\alpha = ({\alpha _1}, \ldots ,{\alpha _n})$ is multi-index, $0 < p < \infty ,0 < a < \infty$ and $- 1 < b < \infty$. These are a generalization of some classical results of Hardy and Littlewood. Using these two inequalities, we generalize a theorem in $[9]$. The methods used in the proof of Theorem 1 lead us to obtain Theorem 7, which enables us to generalize some theorems about the pluriharmonic conjugates in $[8]$ and $[2]$.