Estimates for the integral means of holomorphic functions on bounded domains in ${\displaystyle {ℂ}^n}$

Let = {z ∈ ℂ^{n} : λ(z) < 0} be a bounded domain with ${\displaystyle C^{\infty }}$ boundary. For f holomorphic in , let ${\displaystyle M_p(f,r)}$ be the pth integral mean of f on ${\displaystyle {\partial }_r=\{z\in :\lambda \left(z\right)=-r\}}$. In this paper we prove that ${\displaystyle {\int }_0^{\epsilon }{r}^{s+|\alpha |q}{M}_p^q(D^{\alpha }f,r)dr\le C{\int }_0^{\epsilon }{r}^s{M}_p^q(f,r)dr}$ and ${\displaystyle {\int }_0^{\epsilon }{r}^s{M}_p^q(f,r)dr\le C\{{\sum }_{|\alpha |-1,m\in \mathbb{N},\alpha =({\alpha }_1,...,{\alpha }_n)isa\mu <i-\in dex,\text{and}\epsilon >0issmallenough.These\in equalities\ge \ne ralizetheknownres\underline{t}s\in [9,10]ontheunitballof}}$ℂ^{n}!$!. Two applications are given. The methods used in the proof of the inequalities also enable us to obtain some theorems about pluriharmonic functions on .