Let $f:\mathbb{C}^n\to\mathbb{C}$ be an entire holomorphic function of $n$ complex variables. Then its Nevannlinna characteristic equals $$ m_f(r)=\int_{\partial B(r)}\log^+|f(z)|d\eta(z),\quad\forall r>0, $$ where $\partial B(r)$ is the sphere of radius $r$ and $\eta$ is the normalized Euclidean measure on it. In the $n=1$ case one can deduce from Poisson-Jensen formula that the following estimate holds (e.g., Levin p. 13), $$ \log|f(z)|\le\frac{r+|z|}{r-|z|}m_f(r),\quad\forall |z|<r. $$ I am looking for an analogous estimate of the modulus in terms of the Nevanlinna characteristic for $n>1$. But all various Poisson/Jensen/Nevanlinna formulae I have seen so far work with $n$-dimensional real hypersurfaces $\partial D$ (Bergman-Shilov boundaries etc.) rather than with the natural $2n-1$-dimension sphere $\partial B(r)$, and are thus of no use for me.

**Question:** Does there exist a similar estimate for $n>1$, and where can I find it?

To be more precise, I am looking for an estimate of the form $$ \log|f(z)|\le F\left(\frac{\|z\|}r\right)m_f(r),\quad\forall z\in\mathbb{C}^n,\quad\forall r>0\quad\mbox{s.t.}\quad c_1<\frac{\|z\|}r<c_2, $$ $$ F:(c_1,c_2)\to\mathbb{R}_+,\quad c_1,c_2\in\mathbb{R}_+. $$ Here $F$, $c_1$ and $c_2$ may depend on $f$, if it helps. Thank you.