Modulus bounded by Nevanlinna characteristic in several variables

Question

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.

Answer 1

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 $F(t) = G(t)H(t)$.