Maximum modulus principle

A nonconstant holomorphic function on a connected domain cannot attain a local maximum of its modulus at an interior point.

A nonconstant holomorphic function cannot have a positive interior local minimum of its modulus. Wherever it is nonzero, its reciprocal is holomorphic; a positive modulus minimum would give a modulus maximum of the reciprocal, contradicting the maximum modulus principle. Thus if a holomorphic function on a neighborhood of a closed disk has modulus at the center strictly smaller than its minimum boundary modulus, it has a zero inside the disk. Otherwise the Cauchy estimate applied to its reciprocal would give the opposite inequality.

For a holomorphic scalar function on a neighborhood of a closed annulus, $M(r)$ is the maximum modulus on its radius-$r$ circle. The logarithm of its modulus is subharmonic; compare it with the harmonic function affine in $\log |z|$ which takes the logarithms of the two boundary maxima. The maximum principle for subharmonic functions proves the displayed bound. Zero boundary maxima are handled by adding a positive bound and taking its limit. The same bound holds for the norm of a Banach space-valued holomorphic function by applying scalar functionals and the Hahn-Banach theorem.

A maximum-modulus extension for unbounded domains with suitable growth restrictions. In its polynomial-growth vertical-strip form, bounds $|F(a+it)|\ll (1+|t|{)}^u$ and $|F(b+it)|\ll (1+|t|{)}^v$ for a holomorphic function imply the interpolated exponent $((b-\sigma )u+(\sigma -a)v)/(b-a)$. It is the strip-convexity tool used to move subpower zeta bounds to the right of the critical line back onto that line.

If a function is continuous on the closure of a bounded plane domain and holomorphic inside, the compact closure supplies a point of maximum modulus. Unless the function is constant, the maximum modulus principle places every maximum on the boundary.

Let $f$ be bounded and holomorphic on a half-plane and continuous on its closure. A bound $|f|\le M$ on the boundary line propagates throughout the half-plane. Apply the bounded-domain principle to $f(z)z^{-1/n}$ on expanding truncated half-discs, then let $n\to \infty$.