Maximum Modulus Principle

According to the maximum modulus principle, the highest possible value of the modulus of a function defined on a bounded domain can only occur on the domain boundary.

• It is also known as the maximum modulus theorem for complicated analytic functions.
• The function is stable if the function modulus has an absolute maximum value within the domain. 
• It is an important concept used in complex analysis.
• Subsequently, the maximum modulus principle describes the characteristics of an analytic function that is local maximum across a domain. 
• If the function is not stable, the maximum value can only be reached on the border.
• Phragmén–Lindelöf principle, which is an extension to unbounded domains, is an important application of the maximum modulus principle.

Key Terms: Maximum Modulus Principle, Analytic Functions, Bounded Domain, Continuity, Complex Analysis, Absolute Maximum Value, Domain

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Statement of Maximum Modulus Principle

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According to the maximum modulus principle, if f is a form of holomorphic function, then in that case, modulus function f will not return a local maximum value that is not within the domain of f.

• It can be assumed that G ⊂ C here can be considered the series of the complicated number, which can be bounded and connected with the open series. 
• The greatest value of |f(z)| arises on G, not within G, if f is an analytic function determined on G, assuming f is a constant function. 
• Stated differently, if M represents the highest value of |f(z)| across and within G, then |f(z)| < M for each point z inside G until f is stable.

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Maximum Modulus Principle Proof

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According to the theorem's hypothesis, f is a continuous function both inside and outside of G, which means that f is an analytic function both inside and outside of G then G is a bounded complicated domain.

⇒ At some point on or within G, the function f reaches its maximum value M; therefore assert that this point only exists on G and not within G.

= Assume that this maximum value M of f happens at any location z = a and inside G, if at all possible.

= Consequently, for any z in G, max|f(z)| = |f(a)| = M and |f(z)| ≤ M.

Let us consider a circle

\(\Gamma\) with center an inside G. f has to be present at a point z = b, an adjacent point of a within \(\Gamma\) , such that |f(b)| ≤ M since f is constant throughout G.

= Considering that M - ε = |f(b)|, where ε > 0.

⇒ According to the concept of continuity, since f is continuous at b, there exists δ > 0 such that for any arbitrarily selected ε > 0.

⇒ Each instance |z – b| < δ, ||f(z)| – |f(b)|| < ε/2 ……...(1)

= Considering that |f(z)| - |f(b)| ≤ ||f(z)| - |f(b)|| < ε/2

Thus, |f(z)| - |f(b)| < ε/2

Alternatively, M – ε + ε/2 = M – ε/2 = |f(z)| < |f(b)| + ε/2

As a result, |f(z)| < M – ε/2 ……….. (2)

which means that γ: |z – b| < δ, for every point z within the circle γ having centre b and radius δ.

⇒ Construct a second circle, \(\Gamma\), starting at a and going through b. The circle γ contains the arc PQ of the circle \(\Gamma\)', which means

⇒ When z is on arc PQ, |f(z)| < M – ε/2.

= However, on PQ's primary arc, |f(z)| ≤ M

Let us assume that the circle \(\Gamma\): |b – a| = r. Cauchy's Integral Formula gives us

\(f(a)=\frac{1}{2\pi i} \int_{\Gamma}\frac{f(z)}{z-a}dz\)

⇒ We have \(\Gamma\) we have z – a = \(re^{\Gamma \theta}\), meaning that

\(f(a)=\frac{1}{2\pi i} \int^{2\pi}_{0}\frac{f(a+re^{i\theta})}{re^{i\theta}}re^{i\theta}d\theta\)

\(\frac{1}{2\pi i}\int^{2\pi}_0f(a+re^{i\theta})d\theta\)

⇒ If c is the angle that the arc PQ at center a contains, then

\(f(a)=\frac{1}{2\pi i} \int^{\alpha}_{0}f(a+re^{i\theta})d\theta+\frac{1}{2\pi i}\int^{2\pi}_{\alpha}f(a+re^{i\theta})d\theta\)

Thus,

\(|f(a)|\leq\frac{1}{2\pi i} \int^{\alpha}_{0}|f(a+re^{i\theta})|d\theta+\frac{1}{2\pi i}\int^{2\pi}_{\alpha}|f(a+re^{i\theta}|)d\theta\)

\(< \frac{1}{2\pi i}\int^{\alpha}_0|M-\frac{e}{2}|d\theta+\frac{1}{2\pi i} \int^{2 \pi}_{\alpha}|M|d\theta \)

\(<\frac{\alpha}{2\pi i}(M-\frac{e}{2})+\frac{M}{2\pi i}(2\pi-\alpha)\)

\(=M-\frac{\alpha \epsilon}{2 \pi}\)

Thus the theorem shows that |f(a)| = M < M – \(\frac{\alpha \epsilon}{2 \pi}\) is incompatible. As a result, it is incorrect that we assumed f would reach its maximum value inside G. Therefore, rather than inside G, f must reach its greatest value on G.

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Maximum Modulus Principle Examples

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The example of maximum modulus principle is as follows:

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Things to Remember

• A non-constant analytic function on a connected open set G ⊂ C cannot have a local maximum point in G, as per the maximum modulus principle.
• The open map theorem, which says that a non constant holomorphic function maps open ranges to open collections, can be seen as a specific example of the maximum modulus principle.
• An adequately small open neighborhood of a cannot have an open image if |f| reaches a local maximum at a. F is constant as a result.
• Unless f is constant, |f(z)| < M for each point z within G if M is the greatest value of |f(z)| on and within G.
• The Borel–Carathéodory theorem, an application of the maximum modulus principle, is used to express an analytic function in terms of its real part.
• Another example of a theorem including Hadamard three-lines theorem shows the behaviour of holomorphic functions between two other parallel lines in the complex plane.

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