# Modulus bounded by Nevanlinna characteristic in several variables

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), $$\log|f(z)|\le\frac{r+|z|}{r-|z|}m_f(r),\quad\forall |z| I am looking for an analogous estimate of the modulus in terms of the Nevanlinna characteristic for $$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 $$\log|f(z)|\le F\left(\frac{\|z\|}r\right)m_f(r),\quad\forall z\in\mathbb{C}^n,\quad\forall r>0\quad\mbox{s.t.}\quad c_1<\frac{\|z\|}r $$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.

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