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 $\mathbb{C}^n$?

  • $\begingroup$ 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. $\endgroup$ Nov 25 '15 at 14:49
  • $\begingroup$ 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 $f$)? $\endgroup$
    – rudolph
    Nov 25 '15 at 15:13
  • $\begingroup$ @rudolf: The measure is $\Delta\log|f|$ where $\Delta$ is the Laplace operator. $\endgroup$ Nov 26 '15 at 18:29

See, for example, L. Ronkin, Introduction to the theory of entire functions of several variables, AMS 1974, MR0346175 Ch. I section 3, and Ch. IV, section 4. Other books on entire functions of several variables also contain this, for example Lelong, Gruman, Entire functions of several complex variables, Springer 1986. These are about analytic functions.

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


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