Maximum modulus principle

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In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus |f | cannot exhibit a true local maximum within the domain of f. In other words, either f is a constant function, or, for any point z₀ inside the domain of f there exist other points arbitrarily close to z₀ at which |f | takes larger values.

This entails for example the consequence that if f is defined and holomorphic in the closed unit disk D, the maximum of |f | (which is certainly defined and attained somewhere because |f | is continuous and the closed disk is a compact space) is attained on the unit circle C.

Mathematically, this can be formulated as follows. Let f be defined on some connected open subset D of the complex plane C. If z₀ is a point in D such that

$|f(z_0)|\ge |f(z)|$

for all z in a neighborhood of z₀, then the function f is constant on D.

1 Sketch of the proof
2 Applications
3 References
4 External links

[edit] Sketch of the proof

One uses the equality

log f(z) = log |f(z)| + i arg f(z)

for complex natural logarithms to deduce that log |f(z)| is a harmonic function. Since z₀ 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'(z)=0, and thus that f(z) is constant as well.

By switching to the reciprocal, we can get the minimum modulus principle. It states that if f is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points, then the modulus |f (z)| takes its minimum value on the boundary of D.

Alternatively, the open mapping theorem, which states that a holomorphic function maps open sets to open sets, can be used to give a proof.

[edit] Applications

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

The fundamental theorem of algebra, as may be seen in the classic text "Introduction to Complex Analysis", by Nevanlinna and Paatero.
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.

[edit] References

E.C. Titchmarsh, The Theory of Functions (2nd Ed) (1939) Oxford University Press. (See chapter 5.)
E.D. Solomentsev, "Maximum-modulus principle" SpringerLink Encyclopaedia of Mathematics (2001)