Liouville, Morera, maximum modulus principle

In one sentence

Holomorphic functions are rigid: bounded on all of $\mathbb{C}$ forces constant, a maximum of $|f|$ inside forces constant, and the converse of Cauchy's theorem holds under a hypothesis weak enough to be useful.

Why the exam asks it

Because each theorem has a hypothesis that sounds like a formality and is not — "entire", "bounded on $\mathbb{C}$", "on a domain", "continuous" — and because Liouville is routinely misapplied to functions bounded only on the real line. $\sin z$ is the standard trap and it is set most years in some form.

The idea

Liouville

A bounded entire function is constant.

From Cauchy's estimate with $n = 1$: $|f'(a)| \le M/r$ for every $r$, and letting $r \to \infty$ gives $f' \equiv 0$.

Both words carry weight. Entire means holomorphic on all of $\mathbb{C}$ — $1/z$ is bounded outside the unit disc and is not entire. Bounded means bounded on $\mathbb{C}$, not on some subset.

$\sin z$ is entire, satisfies $|\sin x| \le 1$ for every real $x$, and is not constant. There is no contradiction: on the imaginary axis $\sin(iy) = i\sinh y$, which grows like $e^{y}/2$. At $z = 5i$ the modulus already exceeds $74$. Bounded on $\mathbb{R}$ is not bounded on $\mathbb{C}$, and the option will rely on you not noticing which is claimed.

The useful strengthenings:

• $|f(z)| \le M|z|^{k}$ for large $|z|$ and $f$ entire $\Rightarrow$ $f$ is a polynomial of degree at most $k$. Cauchy's estimate on $f^{(k+1)}$.
• $\operatorname{Re}f$ bounded above $\Rightarrow$ $f$ constant. Apply Liouville to $e^{f}$, using $|e^{f}| = e^{\operatorname{Re}f}$.
• $f$ entire and non-constant $\Rightarrow$ $f(\mathbb{C})$ is dense in $\mathbb{C}$. That is the little Picard theorem's weaker cousin and follows from Liouville applied to $1/(f-w)$.

Fundamental theorem of algebra: if a non-constant polynomial $p$ had no zero, $1/p$ would be entire and bounded — it tends to $0$ at infinity — hence constant. Contradiction.

Morera

If $f$ is continuous on a domain $D$ and $\oint_{\partial T} f = 0$ for every triangle $T \subset D$, then $f$ is holomorphic on $D$.

The converse of Cauchy's theorem, and the hypotheses are exactly what stop it being trivially false. Continuity is required. Vanishing on every triangle is required — one contour, or one family of contours, is not enough. Its usual role is proving that a limit of holomorphic functions is holomorphic, since uniform convergence on compact sets passes through the integral.

Maximum modulus

If $f$ is holomorphic and non-constant on a domain, $|f|$ has no local maximum in that domain.

So on a bounded domain with $f$ continuous up to the boundary, $\max|f|$ is attained on the boundary. On $|z| \le 2$, $|z^2|$ reaches $4$ on the rim and nowhere inside.

Two corollaries worth having:

• Minimum modulus: $|f|$ has no interior minimum unless $f$ has a zero there. The exception is the whole content — apply the maximum principle to $1/f$, which needs $f \ne 0$.
• $\operatorname{Re}f$ and $\operatorname{Im}f$ obey the same principle, via $e^{f}$ and $e^{if}$.

The mechanism is the open mapping theorem: a non-constant holomorphic function maps open sets to open sets, so the image of a neighbourhood of $a$ is a neighbourhood of $f(a)$ and contains points of larger modulus. Nothing can be a local maximum.

Schwarz's lemma

$f : \mathbb{D} \to \mathbb{D}$ holomorphic with $f(0) = 0$. Then $|f(z)| \le |z|$ and $|f'(0)| \le 1$; equality in either, at a single point, forces $f(z) = cz$ with $|c| = 1$.

Proof is the maximum principle applied to $f(z)/z$, extended holomorphically across $0$. The rigidity — one equality anywhere pins the whole function — is the examinable part.

Where intuition breaks

"$f$ is entire and $|f| \le 1$ on $\mathbb{R}$, so $f$ is constant." $\sin z$. Liouville needs bounded on $\mathbb{C}$. The single most reliable trap in this topic.

"$f$ is bounded and holomorphic, so constant." Needs entire. $f(z) = 1/z$ on $|z| > 1$ is bounded and holomorphic and not constant; so is any bounded holomorphic function on the disc.

"$f$ is entire and non-constant, so $|f| \to \infty$." Only for polynomials. $e^{z}$ is entire, non-constant, and tends to $0$ along the negative real axis. What Liouville gives is unboundedness, not divergence.

"A bounded harmonic function on $\mathbb{R}^2$ need not be constant." It is constant — the harmonic Liouville theorem. True, and worth recognising among false options.

"$|f|$ attains its minimum on the boundary." Only if $f$ has no zero inside. $f(z) = z$ on $|z| \le 1$ attains its minimum at the centre.

"Morera needs $\oint f = 0$ for every closed contour." Triangles suffice, which is what makes it usable. And continuity is required — options drop it.

"$\oint_\gamma f = 0$ for one contour, so $f$ is holomorphic." Not Morera. $\overline{z}$ integrates to $0$ round any contour not enclosing the origin and is nowhere holomorphic.

"Schwarz's lemma applies to any map of the disc to itself." It needs $f(0) = 0$. Without that fixed point the correct statement is the Schwarz–Pick inequality.

The exam's angle

1. When Liouville is invoked, check where the bound holds. On $\mathbb{R}$, on a half-plane, on a strip — none of these is $\mathbb{C}$, and $\sin z$ or $e^{z}$ breaks each.
2. Growth conditions mean Cauchy estimates. $|f| \le M|z|^k$ gives a polynomial of degree $\le k$; count the degree from the exponent.
3. A bounded real part means consider $e^{f}$. This trick answers a whole family of options.
4. For maximum modulus, look at the boundary and nowhere else — unless the question is about a minimum, in which case first ask whether $f$ has a zero.
5. If a question says a limit of holomorphic functions is holomorphic, the tool is Morera, with uniform convergence on compact sets.

The night before

• Liouville: bounded entire $\Rightarrow$ constant. $\sin z$ is bounded on $\mathbb{R}$ and unbounded on $\mathbb{C}$ — $\sin(iy) = i\sinh y$.
• $|f| \le M|z|^k$ and entire $\Rightarrow$ polynomial of degree $\le k$.
• $\operatorname{Re}f$ bounded $\Rightarrow$ constant, via $e^{f}$ and $|e^{f}| = e^{\operatorname{Re}f}$.
• Fundamental theorem of algebra from Liouville applied to $1/p$.
• Morera: continuous, and $\oint = 0$ on every triangle $\Rightarrow$ holomorphic. Used for limits of holomorphic functions.
• Maximum modulus: no interior maximum for non-constant $f$; the maximum is on the boundary.
• Minimum modulus: no interior minimum unless $f$ vanishes there. $f(z) = z$ on the disc.
• Open mapping: non-constant holomorphic maps open to open. That is why no interior maximum exists.
• Schwarz: $f:\mathbb{D}\to\mathbb{D}$, $f(0)=0$ $\Rightarrow$ $|f(z)|\le|z|$, $|f'(0)|\le1$; equality forces a rotation.
• Entire and non-constant gives unbounded, not $|f| \to \infty$: $e^z$.