Schottky's and Bloch's theorems

Intuition [Beginner]

Schottky's theorem answers a natural question: how large can a holomorphic function grow if it is forbidden from taking the values $0$ and $1$?

Without any restriction, a holomorphic function on the unit disk can grow as fast as it wants. But omitting two specific values creates a ceiling. Schottky proved that $|f(z)|$ cannot exceed $\exp (C/(1-|z|{)}^{1/2})$ for a universal constant $C$. The closer $z$ gets to the boundary, the faster $f$ can grow — but only polynomially fast in $1/(1-|z|)$, not arbitrarily.

Bloch's theorem answers a different question: how large is the image of a holomorphic function on the unit disk? Bloch proved that if $f$ is holomorphic on the unit disk with $f^{\prime }(0)=1$, then $f(\mathbb{D})$ contains a disk of definite radius $B>0$. The exact value of $B$ (the Bloch constant) is unknown, but the theorem guarantees it is positive.

Both theorems constrain holomorphic functions from below (Bloch) and above (Schottky).

Visual [Beginner]

Two diagrams side by side. Left: the unit disk with a function $f$ mapping into $\mathbb{C}\setminus \{0,1\}$. The function values stay bounded by a curve that blows up near the boundary. Right: the unit disk mapped by $f$ with $f^{\prime }(0)=1$; the image contains a shaded disk of radius $B$.

Schottky puts a ceiling on growth when values are omitted. Bloch puts a floor on the image when the derivative is normalised.

Worked example [Beginner]

Consider $f(z)=e^z$ on the unit disk $|z|<1$. This function omits $0$ (since $e^z\ne 0$) but does not omit $1$ (since $e^0=1$). So Schottky does not apply directly.

Instead consider $g(z)=e^{e^z}$. This omits both $0$ and $1$: $e^{e^z}$ is an exponential (never zero) and $e^{e^z}=1$ only when $e^z=2\pi ik$ for integer $k$, but $e^z$ is real and positive on the real axis. On the unit disk, $|g(z)|\le e^e$, which is finite — consistent with Schottky's bound.

For Bloch: take $f(z)=z+z^2/2$. Then $f^{\prime }(0)=1$. Bloch guarantees the image contains a disk of radius $B$. By direct computation, $f$ maps the unit disk conformally onto a region that contains a disk of radius approximately $0.43$, which is above the known lower bound for $B$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Theorem (Schottky, 1904). For each $M>0$ there exists a constant $C(M)>0$ such that if $f$ is holomorphic on $\mathbb{D}$, $f$ omits $0$ and $1$, and $|f(0)|\le M$, then $|f(z)|\le \exp (C(M)(1-|z|{)}^{-1/2})$ for all $z\in \mathbb{D}$.

The key feature: the bound depends on $|f(0)|$ and on the distance to the boundary, but not on $f$ itself. Every function omitting $0$ and $1$ with bounded initial value satisfies the same growth estimate.

Theorem (Bloch, 1925). There exists a constant $B>0$ (the Bloch constant) such that if $f$ is holomorphic on $\mathbb{D}$ and $f^{\prime }(0)=1$, then $f(\mathbb{D})$ contains a disk of radius $B$.

The best known bounds are $\frac{\sqrt{3}}{4}\approx 0.4330\ge B>\frac{\sqrt{3}}{4}\Gamma (1/3)\Gamma (11/12)/\Gamma (1/4{)}^2\approx 0.4332\cdot 0.4719\approx 0.2045$. The exact value remains an open problem.

Key theorem with proof [Intermediate+]

Theorem (Schottky, sharp form). Let $f$ be holomorphic on $\mathbb{D}$ and omit $0$ and $1$. Then ${\log }^{+}|f(z)|\le C_1{\log }^{+}|f(0)|+C_2+\frac{1}{2}\log \frac{1}{1-|z|}$ for universal constants $C_1,C_2$.

Proof outline. The proof uses a remarkable iteration technique.

Step 1: Lift to the covering space. Since $f$ omits $0$ and $1$, the image lies in $\mathbb{C}\setminus \{0,1\}$. The universal cover of this space is the upper half-plane $\mathbb{H}$ via the modular function $\lambda$. By the monodromy theorem, $f$ lifts: $f=\lambda \circ h$ for some holomorphic $h:\mathbb{D}\to \mathbb{H}$.

Step 2: Schwarz lemma on the lift. For $h:\mathbb{D}\to \mathbb{H}$, compose with a conformal map $\varphi :\mathbb{H}\to \mathbb{D}$ (e.g., $\varphi (w)=(w-i)/(w+i)$). Then $g=\varphi \circ h:\mathbb{D}\to \mathbb{D}$, and by the Schwarz lemma, $|g^{\prime }(0)|\le 1-|g(0){|}^2$.

Step 3: Translate back. The bound on $|g(z)|$ gives a bound on $|h(z)|$, which in turn bounds $|f(z)|=|\lambda (h(z))|$. Since $\lambda$ grows at most polynomially in the height of its argument in $\mathbb{H}$, the bound on $h$ translates to a bound on $f$.

Step 4: Iteration for sharpness. The $\frac{1}{2}$-exponent is obtained by iterating the lifting procedure and using the geometry of the fundamental domain of $\lambda$. $\square$

Bridge. Schottky's theorem builds toward Picard's little theorem 06.01.20 by providing the growth bound that makes the Liouville argument work; the central insight is that omitting two values forces the function to lift into the hyperbolic upper half-plane, where the Schwarz lemma controls growth. This pattern appears again in Nevanlinna theory where the First Main Theorem bounds the proximity function by the characteristic function. The bridge is that value-omission constrains growth via the geometry of the universal cover — Schottky makes this quantitative.

Exercises [Intermediate+]

Advanced results [Master]

Landau's theorem (1904). If $f(z)=a_0+a_{1}z+a_2{z}^2+\cdots$ is holomorphic on $\mathbb{D}$ and omits $0$ and $1$, then $|a_1|\le \Phi (|a_0|)$ for an explicit function $\Phi$. Equivalently, if $f$ is holomorphic on $\mathbb{D}$ with $f(0)=0$, $f^{\prime }(0)=1$, and $f$ omits a single value $a$, then the radius of the schlicht disk of $f$ is bounded below by $R(a)>0$ depending only on $a$.

The Ahlfors-Shimizu characteristic. Ahlfors (1935) gave a geometric interpretation of Nevanlinna theory using the spherical metric, showing that Schottky's theorem is equivalent to a bound on the Ahlfors-Shimizu characteristic $\hat{T}(r,f)={\int }_0^r\frac{A(t)}{t}dt$ where $A(t)$ is the spherical area of $f$ on $|z|<t$.

The Bonk-Chen theorem (2000) gives a sharp lower bound for the Bloch constant: $B\ge \frac{\sqrt{3}}{4}+10^{-14}$, using extremal methods. The conjectured exact value $B=\frac{\sqrt{3}}{4}$ (the Ahlfors conjecture) remains open.

Synthesis. Schottky and Bloch are the two pillars supporting Picard's theorems; Schottky controls growth from above (functions omitting values cannot grow too fast), while Bloch controls range from below (functions with normalised derivative cover a definite disk). The central insight is that both results flow from the hyperbolic geometry of $\mathbb{C}\setminus \{0,1\}$: the Schwarz lemma on the upper half-plane propagates curvature bounds into function-theoretic bounds. This pattern recurs throughout geometric function theory, where the Ahlfors lemma and its generalisations convert curvature assumptions into analytic conclusions. The bridge is that Schottky and Bloch together identify the precise threshold: omitting two values imposes enough curvature to constrain growth, while the universal cover provides the geometric engine.

Full proof set [Master]

Proposition (Bloch's theorem). There exists $B>0$ such that if $f$ is holomorphic on $\mathbb{D}$ with $f^{\prime }(0)=1$, then $f(\mathbb{D})$ contains an open disk of radius $B$.

Proof. Define $b(r)=r(1-r){\max }_{|z|\le r}|f^{\prime }(z)|$ for $0\le r\le 1$. Then $b$ is continuous, $b(0)=0$, and $b(1)=0$. By the maximum of $b$, there exists $r_0$ where $b$ attains its maximum $M$.

Let $z_0$ be a point where $|f^{\prime }(z_0)|={\max }_{|z|\le r_0}|f^{\prime }(z)|$. Then $|f^{\prime }(z_0)|=M/(r_0(1-r_0))$. The disk $|z-z_0|<(1-r_0)/2$ lies inside $\mathbb{D}$. On this disk, by Cauchy's estimate for $f^{\prime \prime }$, one shows that $f$ maps a neighbourhood of $z_0$ onto a disk of radius at least $cM$ for a universal constant $c$. Since $b(0)=0$ and $b^{\prime }(0)=|f^{\prime }(0)|=1$, we get $M\ge 1/4$, giving $B=c/4>0$. $\square$

Connections [Master]

Picard's little theorem 06.01.20 uses Schottky's growth bound to prove that an entire function omitting two values is bounded (hence constant by Liouville); Schottky is the quantitative engine behind Picard's qualitative statement.

Picard's great theorem 06.01.21 extends the Picard conclusion to essential singularities via normal families; the Schottky bound provides the growth control that makes the normal-family argument work.

Normal families and Montel's theorem 06.01.14 are the functional-analytic framework; Schottky's bound implies that families omitting two values are locally bounded (hence normal), connecting back to Montel.

The Phragmen-Lindelof principle 06.01.22 constrains growth on unbounded domains via boundary data; Schottky constrains growth on bounded domains via value-omission — both are maximum-principle upgrades with different auxiliary conditions.

Bibliography [Master]

@article{schottky1904,
  author = {Schottky, Friedrich},
  title = {{\"U}ber den {P}icardschen Satz und die {B}orelschen Ungleichungen},
  journal = {Sitzungsber. Preuss. Akad. Wiss. Berlin},
  pages = {814--821},
  year = {1904}
}

@article{bloch1925,
  author = {Bloch, Andre},
  title = {Les th{\'e}or{\`e}mes de M. {V}aliron sur les fonctions enti{\`e}res et la th{\'e}orie de la normalit{\'e}},
  journal = {Bull. Soc. Math. France},
  volume = {53},
  pages = {33--56},
  year = {1925}
}

@article{landau1904,
  author = {Landau, Edmund},
  title = {{\"U}ber eine Verallgemeinerung des {P}icardschen Satzes},
  journal = {Sitzungsber. Bayer. Akad. Wiss.},
  pages = {357--376},
  year = {1904}
}

@book{stein-shakarchi-complex,
  author = {Stein, Elias M. and Shakarchi, Rami},
  title = {Complex Analysis},
  series = {Princeton Lectures in Analysis},
  volume = {II},
  publisher = {Princeton University Press},
  year = {2003}
}

Historical & philosophical context [Master]

Friedrich Schottky proved his growth bound in 1904 ^{[Schottky 1904]}, twenty-five years after Picard's theorem. Schottky's contribution was making Picard's qualitative result quantitative: not just "the function is constant," but "here is exactly how fast it can grow if it is not constant."

Andre Bloch proved his theorem in 1925 ^{[Bloch 1925]} while hospitalised in a psychiatric facility, producing several landmark results during this period. Edmund Landau immediately recognised the connection to Picard and published his refinement the same year ^{[Landau 1904]}.

The Bloch constant remains one of the most tantalising open problems in geometric function theory. The Ahlfors conjecture that $B=\sqrt{3}/4$ has resisted proof for nearly a century. The best partial results (Bonk-Chen 2000) come from extremal methods that are geometrically natural but analytically formidable.