Throughout, \(\Omega\subseteq\CC\) is a connected open set.
Bounded and entire
Liouville’s theorem. A bounded entire function is constant (Liouville's theorem).
Two hypotheses imply boundedness of an entire function \(f\):
\(\lim_{z\to\infty} f(z)\) exists. Then \(f\) is bounded outside a large closed disc and continuous on that disc, so bounded on \(\CC\).
\(g(w) \coloneqq f(1/w)\) has a removable singularity at \(w=0\). This is the same condition as the existence of \(\lim_{z\to\infty} f(z)\).
Entire, with polynomial growth
Cauchy’s estimates. If \(f\) is entire and \(\abs{f(z)} \leq C\abs{z}^n\) for all large \(\abs z\), then for fixed \(z_0\) and large \(R\) Cauchy’s inequality on the circle \(C_R\) given by \(\abs{z - z_0} = R\) gives \[ \abs{f^{(n+1)}(z_0)} \leq {(n+1)!\, \norm{f}_{C_R} \over R^{n+1}} \leq {(n+1)!\, C(R+\abs{z_0})^n \over R^{n+1}} \to 0 ,\] so \(f^{(n+1)} \equiv 0\) and \(f\) is a polynomial of degree at most \(n\). Liouville’s theorem is the case \(n=0\). See Cauchy estimates and Liouville.
\(\abs f\) has a local maximum in \(\Omega\)
Maximum modulus principle. If \(\abs f\) has a local maximum at a point of \(\Omega\), then \(f\) is constant (Maximum modulus principle). For \(\Omega\) bounded and \(f\) continuous on \(\overline\Omega\), the maximum of \(\abs f\) over \(\overline\Omega\) is attained on \(\bd\Omega\).
If \(f\) does not vanish on \(\Omega\), applying the maximum modulus principle to \(1/f\) shows that a local minimum of \(\abs f\) in \(\Omega\) forces \(f\) to be constant (Minimum modulus principle). The function \(f(z)=z\) on \(\DD\) has a minimum of \(\abs f\) at its zero \(0\) and is not constant.
The image has empty interior
Open mapping theorem. A nonconstant holomorphic function on \(\Omega\) maps open sets to open sets (Open mapping theorem). So \(f\) is constant if its image has empty interior, in particular if
\(f(\Omega)\) lies in a line or a circle,
\(\abs f\) is constant, so that \(f(\Omega)\) lies in a circle,
\(\Re f\) or \(\Im f\) is constant, so that \(f(\Omega)\) lies in a line, or
\(f(\Omega)\) is a discrete set.
Each of these can also be proved from the Cauchy–Riemann equations.
\(f\) vanishes on a set with a limit point in \(\Omega\)
Identity principle. If the zero set of \(f\) has a limit point in \(\Omega\), then \(f\equiv 0\) (Equivalent conditions for a holomorphic function to vanish identically). A segment, an arc, a convergent sequence of distinct points with limit in \(\Omega\), and a subset of \(\RR\cap\Omega\) of positive measure each have a limit point in \(\Omega\).
Applied to \(\sin^2 z + \cos^2 z - 1\), which vanishes on \(\RR\), the identity principle gives \(\sin^2 z + \cos^2 z = 1\) on \(\CC\). See The identity principle.
\(f'\equiv 0\) on \(\Omega\)
If \(f' \equiv 0\) on the connected open set \(\Omega\), then \(f\) is constant: \(f\) is constant along every segment in \(\Omega\), and \(\Omega\) is polygonally connected. The proofs of Liouville’s theorem and of the polynomial-growth bound end with this step.
\(f\) omits values
Little Picard theorem. A nonconstant entire function omits at most one value of \(\CC\) (Picard's little theorem).
If an entire function \(f\) omits every value in a disc \(D_r(a)\), then \(g\coloneqq 1/(f-a)\) is entire with \(\abs g\leq 1/r\), so \(g\), and hence \(f\), is constant by Liouville’s theorem. Casorati–Weierstrass and Picard gives the corresponding statements near an essential singularity.
Hypotheses
| Given | Theorem | Hypotheses |
|---|---|---|
| a bound on \(\CC\) | Liouville | \(f\) entire |
| growth \(\bigo(\abs z^n)\) | Cauchy estimates | \(f\) entire; the bound for large \(\abs z\) |
| a local maximum of \(\abs f\) | maximum modulus | \(\Omega\) connected |
| a local minimum of \(\abs f\) | minimum modulus | \(\Omega\) connected, \(f\) nonvanishing |
| image in a line or circle | open mapping | \(\Omega\) connected |
| a zero set with a limit point | identity principle | \(\Omega\) connected, the limit point in \(\Omega\) |
| two omitted values | little Picard | \(f\) entire |