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.
Article
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.
Contents
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 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
⋅
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 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
−
4
≤
B
≤
3
−
1
2
⋅
Γ
(
1
3
)
Γ
(
11
12
)
Γ
(
1
4
Article from Wikipedia (CC BY-SA 4.0), where it is maintained by volunteer editors.
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.
