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Let $f$ be a function holomorphic on some connected open subset $D$ of the complex plane $\mathbb{C}$ and taking complex values. The maximum principle states that if $f$ is holomorphic within a bounded domain $D$, continuous up to the boundary of $D$, and non-zero at all points, then $|f(z)|$ takes its minimum value on the boundary of $D$.

Question: How far this principal can be generalized for connected open domain's in several complex variables: $D \subset_o \mathbb{C}^n$? How it can be proved? At first I assumed to proceed via induction and reduce it to one dimensional case which we already know. The problem is that in contrast to one dimensional case there is no Cauchy integral formula available. That's serious problem since in dimension one the maximum principle was an immediate consequence of the Cauchy integral formula. Which strategy can be applied here to prove the higher dimensional maximum principle?

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    $\begingroup$ Cauchy is true in many dimensions locally at least (eg on a small polydisc around the point) so no problem to generalize maximum modulus either $\endgroup$
    – Conrad
    May 12, 2020 at 22:19
  • $\begingroup$ There is a Cauchy integral formula in higher dimensions: en.wikipedia.org/wiki/…, and one can prove the maximum principle in higher dimensions via an induction argument (see math.stackexchange.com/questions/2341307/…). $\endgroup$
    – Alex Nolte
    May 12, 2020 at 22:19

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The maximum modulus principle is an easy consequence of that the function is analytic non-constant, so that for any $a$ there is $v$ such that $f(a+z v)$ is analytic non-constant ie. $f(a+zv) =f(a)+ C z^n+O(z^{n+1})$ which takes values larger than $f(a)$ in a neighborhood. For $f$ holomorphic the analyticity is itself a consequence of the Cauchy integral formula, applied to the variables one by one.

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