Complex Analysics/Maximum Principle

The maximum principle is a statement about holomorphic functions from the Complex Analysis. The magnitude ${\displaystyle |f|}$ of a holomorphic function ${\displaystyle f:G\to \mathbb {C} }$ cannot attain any strict local maxima within the domain of definition. Specifically, it asserts the following statement.

Let ${\displaystyle U\subseteq \mathbb {C} }$ be a domain, and let ${\displaystyle f:U\to \mathbb {C} }$ be holomorphic. If ${\displaystyle |f|}$ has a local maximum in ${\displaystyle U}$, then ${\displaystyle f}$ is constant. If ${\displaystyle U}$ is bounded and ${\displaystyle f}$ can be continuously extended to ${\displaystyle {\bar {U}}}$, then ${\displaystyle f}$ attains its maximum on ${\displaystyle \partial U}$.

To prove this, we require a lemma that locally implies the conclusion.

Let ${\displaystyle G\subseteq \mathbb {C} }$ be open, and ${\displaystyle f:G\to \mathbb {C} }$ be holomorphic. Let ${\displaystyle z_{0}\in G}$ be a local maximum point of ${\displaystyle |f|}$. Then ${\displaystyle f}$ is constant in a neighborhood of ${\displaystyle z_{0}}$.

Let ${\displaystyle r>0}$ be chosen such that ${\displaystyle |f(z)|\leq |f(z_{0})|}$ for all ${\displaystyle z\in {\bar {D}}r(z_{0})\subseteq G}$. The Cauchy's integral formula'gives, for all ${\displaystyle \varepsilon \leq r}$:

${\displaystyle f(z_{0})={\frac {1}{2\pi i}}\int {\partial D_{\varepsilon }(z_{0})}{\frac {f(z)}{z-z_{0}}},dz}$

This allows us to establish the following estimation:

We derive the following estimation:

${\displaystyle {\begin{array}{rl}|f(z_{0})|&={\frac {1}{2\pi }}\left|\int _{\partial D_{\varepsilon }(z_{0})}{\frac {f(z)}{z-z_{0}}}\,dz\right|\\&={\frac {1}{2\pi }}\left|\int _{0}^{2\pi }{\frac {f(z_{0}+\varepsilon e^{it})}{\varepsilon \cdot e^{it}}}\varepsilon \cdot i\cdot e^{it}\,dt\right|\\&\leq {\frac {1}{2\pi }}\int _{0}^{2\pi }|f(z_{0}+\varepsilon \cdot e^{it})|\,dt\\&\leq \sup _{t\in [0,2\pi ]}|f(z_{0}+\varepsilon \cdot e^{it})|\\&\leq |f(z_{0})|\end{array}}}$

It follows that the inequality ${\displaystyle \leq }$ must be an equality chain, implying

${\displaystyle {\begin{array}{rl}|f(z_{0})|=&\int _{0}^{2\pi }|f(z_{0})|\cdot {\frac {1}{2\pi }}dt={\frac {1}{2\pi }}\int _{0}^{2\pi }|f(z_{0}+\varepsilon \cdot e^{it})|\,dt\\\Rightarrow &0=\int _{0}^{2\pi }(|f(z)|-|f(z_{0})|)dt\\\Rightarrow &|f(z)|=|f(z_{0})|\end{array}}}$.

Thus, we establish the constancy of ${\displaystyle |f|}$ using the property:

${\displaystyle |f(z)|=|f(z_{0})|}$ for all ${\displaystyle z\in {\bar {D}}_{r}(z_{0})}$,

i.e., ${\displaystyle |f|}$ is constant on ${\displaystyle D_{r}(z_{0})}$.

If ${\displaystyle |f|={\sqrt {{\mathfrak {Re}}(f)^{2}+{\mathfrak {Im}}(f)^{2}}}}$ is constant on ${\displaystyle D_{r}(z_{0})}$, then ${\displaystyle {\mathfrak {Re}}(f)^{2}+{\mathfrak {Im}}(f)^{2}=c}$ must also be constant, where ${\displaystyle c\in \mathbb {R} }$ is a constant.

Since ${\displaystyle f}$ is holomorphic on ${\displaystyle D_{r}(z_{0})}$, the Cauchy-Riemann-Differential equation'apply:

${\displaystyle u:={\mathfrak {Re}}(f),\ v:={\mathfrak {Im}}(f),\ {\text{and}}\ f(z):=u(x,y)+i\cdot v(x,y)}$,

and the following holds:

${\displaystyle {\frac {\partial u(x,y)^{2}+v(x,y)^{2}}{\partial x}}=0\ {\text{and}}\ {\frac {\partial u(x,y)^{2}+v(x,y)^{2}}{\partial y}}=0}$.

Let ${\displaystyle u_{x}={\frac {\partial u}{\partial x}}}$ and ${\displaystyle u_{y}={\frac {\partial u}{\partial y}}}$. Applying the chain rule to the partial derivatives, we obtain:

${\displaystyle 0=2uu_{x}+2vv_{x}}$ and ${\displaystyle 0=2uu_{y}+2vv_{y}}$.

Using the Cauchy-Riemann-Differential equation', replace the partial derivatives of ${\displaystyle v}$ with those of ${\displaystyle u}$:

${\displaystyle u_{x}=v_{y}}$ and ${\displaystyle u_{y}=-v_{x}}$, leading to:

${\displaystyle 0=2\cdot (uu_{x}-vu_{y})}$ and ${\displaystyle 0=2\cdot (uu_{y}+vu_{x})}$.

Squaring the above equations yields:

${\displaystyle 0=(uu_{x}-vu_{y})^{2}=u^{2}u_{x}^{2}-2uu_{x}\cdot vu_{y}+v^{2}u_{y}^{2}}$,

${\displaystyle 0=(uu_{y}+vu_{x})^{2}=u^{2}u_{y}^{2}+2uu_{y}\cdot vu_{x}+v^{2}u_{x}^{2}}$.

Adding these equations gives:

${\displaystyle 0=u^{2}u_{x}^{2}+v^{2}u_{y}^{2}+u^{2}u_{y}^{2}+v^{2}u_{x}^{2}}$.

Factoring out ${\displaystyle u^{2}}$ and ${\displaystyle v^{2}}$:

${\displaystyle 0=u^{2}(u_{x}^{2}+u_{y}^{2})+v^{2}(u_{x}^{2}+u_{y}^{2})=(u^{2}+v^{2})\cdot (u_{x}^{2}+u_{y}^{2})}$.

Thus,

${\displaystyle 0=u^{2}+v^{2}}$ or ${\displaystyle 0=u_{x}^{2}+u_{y}^{2}}$.

With ${\displaystyle u^{2}=-v^{2}}$, it follows that ${\displaystyle u=v=0}$ since ${\displaystyle u(x,y)}$ and ${\displaystyle v(x,y)}$ are real-valued, implying ${\displaystyle f=u+iv=0}$.

If ${\displaystyle u_{x}^{2}=-u_{y}^{2}}$, then ${\displaystyle u_{x}^{2}=u_{y}^{2}=0}$, and ${\displaystyle u_{x}=u_{y}=0}$. By the Cauchy-Riemann-Differential equation, ${\displaystyle f'=0}$. Thus, ${\displaystyle f}$ is constant on ${\displaystyle D_{r}(z_{0})}$.

Let ${\displaystyle z_{0}\in G}$ be a local maximum point of ${\displaystyle |f|}$ in the domain ${\displaystyle G}$. Define ${\displaystyle V:={z\in G:f(z)=f(z_{0})}}$ as the set of all ${\displaystyle z\in G}$ mapped to ${\displaystyle w:=f(z_{0})\in \mathbb {C} }$ (level set).

Since ${\displaystyle f}$ is continuous, preimages of open sets are open, and preimages of closed sets are closed (in the relative topology of ${\displaystyle G}$). Thus, ${\displaystyle V=f^{-1}({w})}$ is closed in ${\displaystyle G}$.

Using the lemma, ${\displaystyle V}$ can also be represented as a union of open disks, and unions of open sets are open.

Thus, ${\displaystyle V=G}$ due to the connectivity of ${\displaystyle U}$, i.e., ${\displaystyle f}$ is constant.

If ${\displaystyle G}$ is bounded, then ${\displaystyle {\bar {G}}}$ is compact. Therefore, the continuous function ${\displaystyle f}$ attains its maximum on ${\displaystyle {\bar {G}}}$, say at ${\displaystyle z_{0}\in {\bar {G}}}$. If ${\displaystyle z_{0}\in G}$, then ${\displaystyle f}$ is constant on ${\displaystyle G}$ (by the lemma) and hence on ${\displaystyle {\bar {G}}}$, so ${\displaystyle f}$ also attains its maximum on ${\displaystyle \partial G}$. Otherwise, ${\displaystyle z_{0}\in \partial G}$, completing the proof.

• Cauchy-Riemann-Differential equation'
• Application of Cauchy-Riemann Equations

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• Date: 12/26/2024