maximum modulus principle

Let $f$ be a holomorphic function on some connected open subset $D$ of the complex plane $\backslash mathbb\{C\}$ and taking complex values. If $z\_0$ is a point in $D$ such that

:$|f(z\_0)|\backslash ge\; |f(z)|$

for all $z$ in some neighborhood of $z\_0$, then $f$ is constant on $D$.

This statement can be viewed as a special case of the open mapping theorem, which states that a nonconstant holomorphic function maps open sets to open sets: If $|f|$ attains a local maximum at $z$, then the image of a sufficiently small open neighborhood of $z$ cannot be open, so $f$ is constant.

Suppose that $D$ is a bounded nonempty connected open subset of $\backslash mathbb\{C\}$.

Let $\backslash overline\{D\}$ be the closure of $D$.

Suppose that $f\; \backslash colon\; \backslash overline\{D\}\; \backslash to\; \backslash mathbb\{C\}$ is a continuous function that is holomorphic on $D$.

Then $|f(z)|$ attains a maximum at some point of the boundary of $D$.

This follows from the first version as follows. Since $\backslash overline\{D\}$ is compact and nonempty, the continuous function $|f(z)|$ attains a maximum at some point $z\_0$ of $\backslash overline\{D\}$. If $z\_0$ is not on the boundary, then the maximum modulus principle implies that $f$ is constant, so $|f(z)|$ also attains the same maximum at any point of the boundary.

For a holomorphic function $f$ on a connected open set $D$ of $\backslash mathbb\{C\}$, if $z\_0$ is a point in $D$ such that

:

for all $z$ in some neighborhood of $z\_0$, then $f$ is constant on $D$.

Proof: Apply the maximum modulus principle to $1/f$.

One can use the equality

:$\backslash log\; f(z)\; =\; \backslash ln\; |f(z)|\; +\; i\backslash arg\; f(z)$

for complex natural logarithms to deduce that $\backslash ln\; |f\; (z)\; |$ is a harmonic function. Since $z\_0$ is a local maximum for this function also, it follows from the maximum principle that $|\; f\; (z)\; |$ is constant. Then, using the Cauchy–Riemann equations we show that $f\text{'}(z)$ = 0, and thus that $f(z)$ is constant as well. Similar reasoning shows that $|\; f\; (z)\; |$ can only have a local minimum (which necessarily has value 0) at an isolated zero of $f(z)$.

Another proof works by using Gauss's mean value theorem to "force" all points within overlapping open disks to assume the same value as the maximum. The disks are laid such that their centers form a polygonal path from the value where $f(z)$ is maximized to any other point in the domain, while being totally contained within the domain. Thus the existence of a maximum value implies that all the values in the domain are the same, thus $f(z)$ is constant.

Source:{{cite book |last1=Conway |first1=John B. |editor1-last=Axler |editor1-first=S. |editor2-last=Gehring |editor2-first=F.W. |editor3-last=Ribet |editor3-first=K.A. |title=Functions of One Complex Variable I |date=1978 |publisher=Springer Science+Business Media, Inc. |location=New York |isbn=978-1-4612-6314-2 |edition=2}}

As $D$ is open, there exists $\backslash overline\{B\}(a,r)$ (a closed ball centered at $a\; \backslash in\; D$ with radius $r>0$) such that $\backslash overline\{B\}(a,r)\; \backslash subset\; D$. We then define the boundary of the closed ball with positive orientation as $\backslash gamma(t)=a+re^\{it\},\; t\; \backslash in\; [0,2\backslash pi]$. Invoking Cauchy's integral formula, we obtain

:$0\; \backslash leq\; \backslash int\_\{0\}^\{2\backslash pi\}\; |f(a)|-|\; f(a+re^\{it\})|\; \backslash ,dt\; \backslash leq\; 0$

For all $t\; \backslash in\; [0,2\backslash pi]$, $|\; f(a)\; |-|\; f(a+re^\{it\})\; |\; \backslash geq\; 0$, so $|\; f(a)|=|\; f(a+re^\{it\})\; |$. This also holds for all balls of radius less than $r$ centered at $a$. Therefore, $f(z)=f(a)$ for all $z\; \backslash in\; \backslash overline\{B\}(a,r)$.

Now consider the constant function $g(z)=f(a)$ for all $z\; \backslash in\; D$. Then one can construct a sequence of distinct points located in $\backslash overline\{B\}(a,r)$ where the holomorphic function $g-f$ vanishes. As $\backslash overline\{B\}(a,r)$ is closed, the sequence converges to some point in $\backslash overline\{B\}(a,r)\; \backslash in\; D$. This means $f-g$ vanishes everywhere in $D$ which implies $f(z)=f(a)$ for all $z\; \backslash in\; D$.

A physical interpretation of this principle comes from the heat equation. That is, since $\backslash log\; |\; f(z)\; |$ is harmonic, it is thus the steady state of a heat flow on the region $D$. Suppose a strict maximum was attained on the interior of $D$, the heat at this maximum would be dispersing to the points around it, which would contradict the assumption that this represents the steady state of a system.

The maximum modulus principle has many uses in complex analysis, and may be used to prove the following:

• The fundamental theorem of algebra.
• Schwarz's lemma, a result which in turn has many generalisations and applications in complex analysis.
• The Phragmén–Lindelöf principle, an extension to unbounded domains.
• The Borel–Carathéodory theorem, which bounds an analytic function in terms of its real part.
• The Hadamard three-lines theorem, a result about the behaviour of bounded holomorphic functions on a line between two other parallel lines in the complex plane.

{{Reflist}}

• {{cite book |first=E. C. |last=Titchmarsh |authorlink=Edward Charles Titchmarsh |title=The Theory of Functions |url=https://archive.org/details/in.ernet.dli.2015.2588 |edition=2nd |year=1939 |publisher=Oxford University Press }} (See chapter 5.)
• {{springerEOM|author=E. D. Solomentsev|title=Maximum-modulus principle|id=m/m063110}}
• Conway, John B. (1978). Axler, S.; Gehring, F.W.; Ribet, K.A. (eds.). Functions of One Complex Variable I (2 ed.). New York: Springer Science+Business Media, Inc. ISBN 978-1-4612-6314-2.

• {{MathWorld | urlname= MaximumModulusPrinciple | title= Maximum Modulus Principle}}

Category:Mathematical principles

Category:Theorems in complex analysis