# Bloch's theorem (complex analysis)

In complex analysis, a branch of mathematics, Bloch's theorem describes the behaviour of holomorphic functions defined on the unit disk. It gives a lower bound on the size of a disk in which an inverse to a holomorphic function exists. It is named after André Bloch.

## Statement

Let f be a holomorphic function in the unit disk |z| ≤ 1 for which
    $|f'(0)|=1$
Bloch's theorem states that there is a disk S ⊂ D on which f is biholomorphic and f(S) contains a disk with radius 1/72.

## Landau's theorem

If f is a holomorphic function in the unit disk with the property |f′(0)| = 1, then let L$_f$ be the radius of the largest disk contained in the image of f.
Landau's theorem states that there is a constant L defined as the infimum of L$_f$ over all such functions f, and that L is greater than Bloch's constant L ≥ B.
This theorem is named after Edmund Landau.

## Valiron's theorem

Bloch's theorem was inspired by the following theorem of Georges Valiron:
Theorem. If f is a non-constant entire function then there exist disks D of arbitrarily large radius and analytic functions φ in D such that f(φ(z)) = z for z in D.
Bloch's theorem corresponds to Valiron's theorem via the so-called Bloch's principle.

## Proof
### Landau's theorem

We first prove the case when f(0) = 0, f′(0) = 1, and |f′(z)| ≤ 2 in the unit disk.
By Cauchy's integral formula, we have a bound
          $|f''(z)|=\left|{\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f'(w)}{(w-z)^{2}}}\,\mathrm {d} w\right|\leq {\frac {1}{2\pi }}\cdot 2\pi r\sup _{w=\gamma (t)}{\frac {|f'(w)|}{|w-z|^{2}}}\leq {\frac {2}{r}},$
where γ is the counterclockwise circle of radius r around z, and 0 < r < 1 − |z|.
By Taylor's theorem, for each z in the unit disk, there exists 0 ≤ t ≤ 1 such that f(z) = z + z$^2$f″(tz) / 2.
Thus, if |z| = 1/3 and |w| < 1/6, we have
          $|(f(z)-w)-(z-w)|={\frac {1}{2}}|z|^{2}|f''(tz)|\leq {\frac {|z|^{2}}{1-t|z|}}\leq {\frac {|z|^{2}}{1-|z|}}={\frac {1}{6}}<|z|-|w|\leq |z-w|.$
By Rouché's theorem, the range of f contains the disk of radius 1/6 around 0.
Let D(z$_0$, r) denote the open disk of radius r around z$_0$. For an analytic function g : D(z$_0$, r) → C such that g(z$_0$) ≠ 0, the case above applied to (g(z$_0$ + rz) − g(z$_0$)) / (rg′(0)) implies that the range of g contains D(g(z$_0$), |g′(0)|r / 6).
For the general case, let f be an analytic function in the unit disk such that |f′(0)| = 1, and z$_0$ = 0.
* If |f′(z)| ≤ 2|f′(z$_0$)| for |z − z$_0$| < 1/4, then by the first case, the range of f contains a disk of radius |f′(z$_0$)| / 24 = 1/24.
* Otherwise, there exists z$_1$ such that |z$_1$ − z$_0$| < 1/4 and |f′(z$_1$)| > 2|f′(z$_0$)|.
* If |f′(z)| ≤ 2|f′(z$_1$)| for |z − z$_1$| < 1/8, then by the first case, the range of f contains a disk of radius |f′(z$_1$)| / 48 > |f′(z$_0$)| / 24 = 1/24.
* Otherwise, there exists z$_2$ such that |z$_2$ − z$_1$| < 1/8 and |f′(z$_2$)| > 2|f′(z$_1$)|.
Repeating this argument, we either find a disk of radius at least 1/24 in the range of f, proving the theorem, or find an infinite sequence (z$_n$) such that |z$_n$ − z$_{n−1}$| < 1/2$^{n+1}$ and |f′(z$_n$)| > 2|f′(z$_{n−1}$)|.
In the latter case the sequence is in D(0, 1/2), so f′ is unbounded in D(0, 1/2), a contradiction.

### Bloch's theorem

In the proof of Landau's Theorem above, Rouché's theorem implies that not only can we find a disk D of radius at least 1/24 in the range of f, but there is also a small disk D$_0$ inside the unit disk such that for every w ∈ D there is a unique z ∈ D$_0$ with f(z) = w. Thus, f is a bijective analytic function from D$_0$ ∩ f$^{−1}$(D) to D, so its inverse φ is also analytic by the inverse function theorem.

## Bloch's and Landau's constants

The number B is called the Bloch's constant. The lower bound 1/72 in Bloch's theorem is not the best possible. Bloch's theorem tells us B ≥ 1/72, but the exact value of B is still unknown.
The best known bounds for B at present are
    $0.4332\approx {\frac {\sqrt {3}}{4}}+2\times 10^{-14}\leq B\leq {\sqrt {\frac {{\sqrt {3}}-1}{2}}}\cdot {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {11}{12}})}{\Gamma ({\frac {1}{4}})}}\approx 0.47186,$
where Γ is the Gamma function. The lower bound was proved by Chen and Gauthier, and the upper bound dates back to Ahlfors and Grunsky.
The similarly defined optimal constant L in Landau's theorem is called the Landau's constant. Its exact value is also unknown, but it is known that
    $0.5<L\leq {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {5}{6}})}{\Gamma ({\frac {1}{6}})}}=0.543258965342...\,\!$ (sequence A081760 in the OEIS)
In their paper, Ahlfors and Grunsky conjectured that their upper bounds are actually the true values of B and L.
For injective holomorphic functions on the unit disk, a constant A can similarly be defined. It is known that
    $0.5<A\leq 0.7853$