The Jensen's formula says the following: Let $f$ be analytic on the disc $D$ of radius $R$ centered at the origin such that $f(0)\neq 0$, then \begin{align} \log(|f(0)|)+ \sum_{i=1}^n \log \left(\frac{R}{a_i} \right ) = \frac{1}{2\pi}\int_0^{2\pi}\log(|f(Re^{i\theta})|)d\theta, \end{align} where $a_i$'s are zerof of $f$ in $D$.

Are there extensions of this theorem to regions other than discs. For example, does it hold of rectangular regions?

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    $\begingroup$ I suppose a good start is with precomposing with some conformal maps of your choice to see how the formula changes. $\endgroup$ Jun 19 '19 at 0:02

Yes, there are. See Theorem 3.13 in

Hayman, W. K.; Kennedy, P. B. Subharmonic functions. Vol. I. (English) Zbl 0419.31001 London Mathematical Society Monographs. No. 9. London-New York-San Francisco: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers, XVII, 284 p. (1976). MSC: 31-02 31A05 31A15 31B05 31B15 31D05 (see the second page of the review for its formulation).


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