Maximum-modulus principle

This principle is also called the maximum principle, see [Bu].

A theorem expressing one of the basic properties of the modulus of an 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 or a 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. 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. 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 zero (in $\mathbb{R}^{2n}=\mathbb{C}^n$), then $\lvert f(z)\rvert\leq M$ everywhere in $D$. See also Two-constants 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 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 $\lVert f\rVert=\sqrt{\lvert f_1\rvert^2+\cdots+\lvert f_m\rvert^2}$ 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 principle of preservation of domain is satisfied.

References

[Ah]	L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241
[Bu]	R.B. Burchel, "An introduction to classical complex analysis" , 1 , Acad. Press (1979)
[Sh]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)
[St]	S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian)
[Vl]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)