Maximum modulus principle

A nonconstant holomorphic function cannot attain a local maximum of its modulus at an interior point of its domain. Its modulus therefore reaches any maximum on the boundary of a bounded domain where the function is continuous.

Mean value property: A function's value at the center of a suitable region equals an average of its values on that region's boundary. Holomorphic functions satisfy this property, which prevents an isolated interior modulus peak.
Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. Holomorphicity supplies the rigidity that rules out a nonconstant interior modulus maximum.
Maximum modulus theorem: The result that a function holomorphic on a bounded domain and continuous on its closure has its maximum modulus on the boundary. This boundary conclusion follows directly from the interior prohibition.
Minimum modulus principle: A nonvanishing holomorphic function cannot attain a local minimum of its modulus inside its domain unless it is constant. It is the corresponding interior-extremum restriction for minima rather than maxima.
Cauchy's integral formula: A formula expressing a holomorphic function's interior values as integrals of its boundary values. It yields the mean-value property used in a standard proof of the principle.
Complex modulus: The nonnegative distance of a complex number from zero in the complex plane. The principle concerns local maxima of this quantity, not maxima of the function's complex values.
Liouville's theorem: Every bounded entire function is constant. Applying the principle on increasingly large disks proves this rigidity result.
Maximum principle: A family of results stating that interior extrema of solutions are constrained by boundary values or force constancy. The maximum modulus principle is the complex-analytic member of this broader family.
Taylor series: A power series expansion of a function around a point, with coefficients determined by its derivatives there. The first nonzero higher-order coefficient creates nearby values whose modulus exceeds a proposed local maximum.
Local maximum: A value at a point that is at least as large as all nearby values of a function. The theorem forbids this condition for the modulus at an interior point of a nonconstant holomorphic function.

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