, the maximum modulus principle
in complex analysis
states that if f
is a holomorphic function
, then the modulus
cannot exhibit a true local maximum
that is properly within the domain
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.
be a function holomorphic on some connected open subset D
of the complex plane C
and taking complex values. If z0
is a point in D
for all z
in a neighborhood
, then the function f
is constant on D
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
)| is a harmonic function
. Since z0
is a local maximum for this function also, it follows from the maximum principle
)| is constant. Then, using the Cauchy-Riemann equations
we show that f
)=0, and thus that f
) 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 maximum modulus principle can be viewed as a special case of the open mapping theorem, which states that a holomorphic function maps open sets to open sets. If |f| attains a local maximum at a, then clearly the direct image of sufficiently small open neighborhoods of a cannot be open. Therefore, f is constant.
The maximum modulus principle has many uses in complex analysis, and may be used to prove the following:
- E.C. Titchmarsh, The Theory of Functions (2nd Ed) (1939) Oxford University Press. (See chapter 5.)