proof of maximal modulus principle

f:U is holomorphic and therefore continuousPlanetmathPlanetmath, so |f| will also be continuous on U. KU is compactPlanetmathPlanetmath and since |f| is continuous on K it must attain a maximum and a minimum value there.

Suppose the maximum of |f| is attained at z0 in the interior of K.

By definition there will exist r>0 such that the set Sr={z:|z-z0|2r2}K.

Consider Cr the boundary of the previous set parameterized counter-clockwise. Since f is holomorphic by hypothesisMathworldPlanetmath, Cauchy integral formulaPlanetmathPlanetmath says that

f(z0)=12πiCf(z)z-z0𝑑z (1)

A canonical parameterization of Cr is z=z0+reiθr, for θ[0,2πr].

f(z0)=12πr02πrf(z0+reiθr)𝑑θ (2)

Taking modulus on both sides and using the estimating theorem of contour integral


Since |f(z0)| is a maximum, the last inequality must be verified by having the equality in the verified.

The proof of the estimating theorem of contour integral ( implies that equality is only verified when


where λ is a constant. Therefore, f(zo+reiθr) is constant and to verify equation 2 its value must be f(z0).

So f is holomorphic and constant on a circumference. It’s a well known result that if 2 holomorphic functions are equal on a curve, then they are equal on their entire domain, so f is constant.

to see this in this particular circumstance is using equation 1 to calculate the value of f on a point ξ interior Sr different than z0. Bearing in mind that f(z)=f(z0) is constant in Cr the formulaMathworldPlanetmath reads f(ξ)=f(z0)2πiCr1z-ξ𝑑z=f(z0). So f is really constant in the interior of Sr and the only holomorphic function defined in K that is constant in the interior of Sr is the constant function on all K.

Thus if the maximum of |f| is attained in the interior of K, then f is constant. If f isn’t constant, the maximum must be attained somewhere in K, but not in its interior. Since K is compact, by definition it must be attained at K.

Title proof of maximal modulus principle
Canonical name ProofOfMaximalModulusPrinciple
Date of creation 2013-03-22 15:46:15
Last modified on 2013-03-22 15:46:15
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 19
Author cvalente (11260)
Entry type Proof
Classification msc 30F15
Classification msc 31B05
Classification msc 31A05
Classification msc 30C80