Maximum modulus principle
Theorem. If \(f\) is holomorphic and nonconstant on a connected domain \(U\), then \(|f|\) has no local maximum in \(U\). If \(|f|\) attains an interior maximum, the mean value property forces \(f\) to be constant on a small disc and hence everywhere by identity.
Minimum principle
Corollary. If \(f\) is holomorphic and nowhere zero, then \(|f|\) has no interior local minimum unless \(f\) is constant. Apply the maximum principle to \(1/f\).
Boundary maximum on compact sets
Corollary. If \(f\) is continuous on \(\overline U\), holomorphic on bounded \(U\), and \(U\) has compact closure, then \(\max_{\overline U}|f|\) is achieved on \(\partial U\), unless \(f\) is constant.
Open mapping theorem
Theorem. A nonconstant holomorphic function maps open sets to open sets. Near a zero of order \(m\), one can write \(f(z)-f(a)=(z-a)^m g(z)\) with \(g(a)\ne0\), and the image contains a small disc around \(f(a)\).
Schwarz-type estimate
Example. If \(|f|\le1\) on the unit disc and \(f(0)=0\), the maximum principle applied to \(f(z)/z\) gives \(|f(z)|\le |z|\). This is the core estimate behind Schwarz's lemma.
Harmonic parallel
Remark. The real and imaginary parts of holomorphic functions are harmonic. The maximum principle for holomorphic functions mirrors the maximum principle for harmonic functions.
Maximum modulus and its strong form
Theorem (maximum modulus). If a holomorphic \(f\) on a domain \(\Omega\) has a local maximum of \(|f|\), then \(f\) is constant.
Proof. Let \(z_0\) be a local maximum and choose a contained circle. The mean-value formula gives \[ f(z_0)=\frac1{2\pi}\int_0^{2\pi}f(z_0+re^{it})\,\mathrm{d}t. \] The triangle inequality and maximality force equality throughout. Equality in the complex triangle inequality implies the boundary values all have the same argument and modulus as \(f(z_0)\), hence are equal. Cauchy's formula then makes \(f\) constant inside the disc; the identity theorem extends constancy over \(\Omega\).
Corollary (minimum modulus). If \(f\) is nonvanishing and \(|f|\) has a local minimum, then \(f\) is constant. Apply the maximum principle to \(1/f\).
Open mapping theorem
Theorem. A nonconstant holomorphic map sends open sets to open sets.
Proof. Fix \(z_0\) and write \[ f(z)-f(z_0)=(z-z_0)^mg(z),\qquad g(z_0)\ne0. \] Choose a small circle on which \(|f(z)-f(z_0)|\ge\delta>0\). For \(|w-f(z_0)|<\delta\), Rouché's theorem shows \(f(z)-w\) has the same number \(m\) of zeros inside as \(f(z)-f(z_0)\). Hence every such \(w\) is in the image. A proof avoiding Rouché uses the minimum principle on \(f-w\).
Boundary maximum and uniqueness
If \(\Omega\) is bounded, \(f\) is continuous on \(\overline\Omega\), and holomorphic inside, then \[ \max_{\overline\Omega}|f|=\max_{\partial\Omega}|f|. \] Applied to differences, this gives uniqueness for many boundary-value problems. For harmonic \(u\), apply the holomorphic maximum principle locally to \(e^{f}\) when a harmonic conjugate exists, or use the harmonic mean-value property directly.
Subharmonic logarithms
Where \(f\ne0\), \(\log|f|\) is harmonic; across zeros it is subharmonic. The maximum principle is therefore part of a broader potential-theoretic fact: a subharmonic function cannot have a strict interior maximum. Jensen's formula later turns the logarithmic singularities at zeros into a quantitative counting identity.