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Bloch's theorem (complex analysis)
Theorem in complex analysis named after André Bloch

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.

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Statement

Let f be a holomorphic function in the unit disk |z| ≤ 1 for which

| f ′ ( 0 ) | = 1 {\displaystyle |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 Lf 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 Lf 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 ) | = | 1 2 π i ∮ γ f ′ ( w ) ( w − z ) 2 d w | ≤ 1 2 π ⋅ 2 π r sup w = γ ( t ) | f ′ ( w ) | | w − z | 2 ≤ 2 r , {\displaystyle |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 + z2f″(tz) / 2.

Thus, if |z| = 1/3 and |w| < 1/6, we have

| ( f ( z ) − w ) − ( z − w ) | = 1 2 | z | 2 | f ″ ( t z ) | ≤ | z | 2 1 − t | z | ≤ | z | 2 1 − | z | = 1 6 < | z | − | w | ≤ | z − w | . {\displaystyle |(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(z0, r) denote the open disk of radius r around z0. For an analytic function g : D(z0, r) → C such that g(z0) ≠ 0, the case above applied to (g(z0 + rz) − g(z0)) / (rg′(0)) implies that the range of g contains D(g(z0), |g′(0)|r / 6).

For the general case, let f be an analytic function in the unit disk such that |f′(0)| = 1, and z0 = 0.

  • If |f′(z)| ≤ 2|f′(z0)| for |z − z0| < 1/4, then by the first case, the range of f contains a disk of radius |f′(z0)| / 24 = 1/24.
  • Otherwise, there exists z1 such that |z1 − z0| < 1/4 and |f′(z1)| > 2|f′(z0)|.
  • If |f′(z)| ≤ 2|f′(z1)| for |z − z1| < 1/8, then by the first case, the range of f contains a disk of radius |f′(z1)| / 48 > |f′(z0)| / 24 = 1/24.
  • Otherwise, there exists z2 such that |z2 − z1| < 1/8 and |f′(z2)| > 2|f′(z1)|.

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 (zn) such that |zn − zn−1| < 1/2n+1 and |f′(zn)| > 2|f′(zn−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 D0 inside the unit disk such that for every w ∈ D there is a unique z ∈ D0 with f(z) = w. Thus, f is a bijective analytic function from D0 ∩ 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 ≈ 3 4 + 2 × 10 − 14 ≤ B ≤ 3 − 1 2 ⋅ Γ ( 1 3 ) Γ ( 11 12 ) Γ ( 1 4 ) ≈ 0.47186 , {\displaystyle 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 ≤ Γ ( 1 3 ) Γ ( 5 6 ) Γ ( 1 6 ) = 0.543258965342... {\displaystyle 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 ≤ 0.7853 {\displaystyle 0.5<A\leq 0.7853}

See also