# Jensen formula in $\mathbb{C}^n$?

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function with zero set $X\subset \mathbb{C}$. Jensen's formula reads $$\log(|f(0)|)+\int_0^R\frac{|X\cap B_t(0)|}{t}dt = \frac{1}{2\pi}\int_0^{2\pi}\log(|f(Re^{i\theta})|)d\theta,$$ where $B_t(0)$ denote the ball of radius $t$ around $0$. This formula for instance allows to bound the density of $X$ in terms of growth of $f$.

My question is: does there exist a similar multivariate generalization to $\mathbb{C}^n$?

• Jensen's formula holds for arbitrary subharmonic functions in any dimension. If $f$ is analytic in $C^n$, then $\log|f|$ is (pluri)-subharmonic, and Jensen's formula applies. Nov 25 '15 at 14:49
• Dear Alexandre: Thank you for your comment. What precisely does it mean that `Jensen's formula holds for arbitrary subharmonic functions in any dimension'? How does the formula look (it has to look different; surely it won't involve counting of zeros, but is there a formula which somehow includes a density measure of the zeroes of $f$)? Nov 25 '15 at 15:13
• @rudolf: The measure is $\Delta\log|f|$ where $\Delta$ is the Laplace operator. Nov 26 '15 at 18:29

For subharmonic functions in $R^n$, a reference is Hayman, Kennedy, Subharmonic functions, vol. I.