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.



Modulus bounded by Nevanlinna characteristic in several variables
I belive the following should work: basically the only thing we know is 2-dimensional estimate that you wrote (I'll always talk about real dimension to avoid ambiguity). Then to get estimate of $\log |f(z\_0)|$ in terms of $\int\_{\partial B(r)} \log^{+}|f(z)|d\eta(z)$ let's take average over all 2-dimensional planes passing through $z\_0$. First of all all sections by these planes are 2-dimensional disks and $z\_0$ is not to close to its boundary, so that we have something like $\log |f(z\_0)| \le G\left(\frac{||z\_0||}{r}\right) m\_{f}(T) $, where $m\_f(T)$ is Nevannlinna characteristic in the section by 2-dimensional plane $T$ (actually, distance from $z\_0$ to the sphere and therefore to the circle is at least $r - ||z\_0||$ while radius of all circles is at most $r$). Finally it is not very hard to convince oneself that when you average all $T$'s every point on the $\partial B(r)$ will contribute to the result with coefficient bounded by some other $H\left(\frac{||z\_0||}{r}\right)$ and we therefore get the desired bound with $F(t) = G(t)H(t)$.


