View source for Maximum-modulus principle

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This principle is also called the maximum principle, see {{Cite|Bu}}.

A theorem expressing one of the basic properties of the modulus of an [[Analytic function|analytic function]]. Let $f(z)$ be a regular analytic, or holomorphic, function of $n$ complex variables $z=(z_1,\ldots,z_n)$, $n\geq1$, defined on an (open) domain $D$ of the complex space $\mathbb{C}^n$, which is not a constant, $f(z)\neq\textrm{const}$. The local formulation of the maximum-modulus principle asserts that the modulus of $f(z)$ does not have a local maximum at a point $z_0\in D$, that is, there is no neighbourhood $V(z_0)$ of $z_0$ for which $\lvert f(z)\rvert\leq\lvert f(z_0\rvert$, $z\in V(z_0)$. If in addition $f(z_0)\neq 0$, then $z_0$ also cannot be a local minimum point of the modulus of $f(z)$. An equivalent global formulation of the maximum-modulus principle is that, under the same conditions as above, the modulus of $f(z)$ does not attain its least upper bound

\[ M=\sup\{ \lvert f(z)\rvert : z\in D\}\]

at any $z_0\in D$. Consequently, if $f(z)$ is continuous in a finite closed domain $D$, then $M$ can only be attained on the boundary of $D$. These formulations of the maximum-modulus principle still hold when $f(z)$ is a holomorphic function on a connected complex (analytic) manifold, in particular, on a [[Riemann surface|Riemann surface]] or a [[Riemannian domain|Riemann domain]] $D$.

The maximum-modulus principle has generalizations in several directions. First, instead of $f(z)$ being holomorphic, it is sufficient to assume that $f(z)=u(z)+iv(z)$ is a (complex) [[Harmonic function|harmonic function]]. Another generalization is connected with the fact that for a holomorphic function $f(z)$ the modulus $\lvert f(z)\rvert$ is a [[Logarithmically-subharmonic function|logarithmically-subharmonic function]]. If $f(z)$ is a bounded holomorphic function in a finite domain $D\subset \mathbb{C}^n$ and if

\[ \limsup \{ \lvert f(z)\rvert : z\to \zeta, z\in D\}\leq M\]

holds for all $\zeta\in\partial D$, except at some set $E\subset \partial D$ of outer [[Capacity|capacity]] zero (in $\mathbb{R}^{2n}=\mathbb{C}^n$), then $\lvert f(z)\rvert\leq M$ everywhere in $D$. See also [[Two-constants theorem|Two-constants theorem]]; [[Phragmén–Lindelöf theorem|Phragmén–Lindelöf theorem]].

The maximum-modulus principle can also be generalized to holomorphic mappings. Let $f: D\to\mathbb{C}^n$ be a [[Holomorphic mapping|holomorphic mapping]] of an (open) domain $D\subset\mathbb{C}^n$, $n\geq 1$, into $\mathbb{C}^m$, that is, $f=(f_1,\ldots, f_m)$, $m\geq 1$, where $f_j$, $j=1,\ldots,m$, are holomorphic functions on $D$, $f(z)\neq\textrm{const}$ and $\lVert f\rVert=\sqrt{\lvert f_1\rvert^2+\cdots+\lvert f(z_m)\rvert^2}$ is the Euclidean norm. Then $\lVert f(z)\rVert$ does not attain a local maximum at any $z_0\in D$. The maximum-modulus principle is valid whenever the [[Preservation of domain, principle of|principle of preservation of domain]] is satisfied.

====References====
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|valign="top"|{{Ref|Ah}}||valign="top"| L.V. Ahlfors,   "Complex analysis" , McGraw-Hill  (1979)  pp. 241
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|valign="top"|{{Ref|Bu}}||valign="top"| R.B. Burchel,   "An introduction to classical complex analysis" , '''1''' , Acad. Press  (1979)
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|valign="top"|{{Ref|Sh}}||valign="top"| B.V. Shabat,   "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)
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|valign="top"|{{Ref|St}}||valign="top"| S. Stoilov,   "The theory of functions of a complex variable" , '''1–2''' , Moscow  (1962)  (In Russian; translated from Rumanian)
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|valign="top"|{{Ref|Vl}}||valign="top"| V.S. Vladimirov,   "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)
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